Physics 2220 Fall 2010

George Williams

THIRD MIDTERM - REVIEW PROBLEMS

Solution sets are available on the course web site. A data sheet is provided. Problems marked by "*" do not have

solutions.

1. An electron is accelerated from rest by a potential difference of 525 volts.

The electron is moving horizontally in a vacuum and enters a region where

there is a magnetic field of 1.32 × 10 T at right angles to its velocity. After-4

it travels 12.5 cm (measured along its actual path) in the magnetic field the

electron strikes a screen. Calculate the deflection of the electron from the

path it would have followed in the absence of the magnetic field.

2. Electrons are incident from the left on a small hole. There is a

uniform electric field of 15750 V/m over the entire region in the

vertical direction, as shown. If the electrons cross the region

from wall to wall in 2.00 × 10 s, calculate the magnetic field-7

necessary for the electron to travel across in a straight line. (A

numerical value, plus a clear statement of direction, is needed.)

3. (a) Take the Earth's magnetic field as being along a N-S direction. What is the direction of the force on a

wire carrying current from E to W?

(b) Calculate the maximum value of the torque on a circular coil of wire carrying 11.0 A, if the coil has

17.0 turns and radius of 1.5 cm. The magnetic field is 1700 gauss.

(c) A capacitor of C = 175 :F charged to 100 V is discharged through a 11,000 S resistor. Calculate the

voltage on the capacitor after 2 time constants have passed (from the beginning of discharge).

(d) Calculate the magnitude of the magnetic flux, in Wb, through the lecture table in 101 JFB. Assume the

table is horizontal. Take the Earth's field as 0.500 gauss, at an angle of 70° from the horizontal. The

table is 1.00 m wide and 6.00 m long.

(e) Calculate the radius of the orbit of an electron of v = 1.2 × 10 m/s, in a magnetic field of 150 gauss, if3

the field is perpendicular to the plane of the orbit.

4. Mercury ions (singly charged mercury atoms) are passed through a velocity

selector, as discussed in class, with E = 125,000 V/m and B = 0.850 T. They

pass through a slit and follow a curved path in a field B = 0.850 T

perpendicular to the paper. Calculate the distance, d, for two mercury isotopes

202 and 200. Take the masses as 202 and 200 times the mass of the proton (not

precisely true).

5. (a) Calculate the cyclotron frequency for electrons in a magnetic field of 0.010 T.

(b) Protons are measured to travel in a circular path of radius 6.00 cm in a magnetic field of 1.50 T.

Calculate their velocity.

(c) If a DC power line is carrying a steady current of 12,500 A in the earth's magnetic field which is

assumed to be 0.500 gauss, what is the maximum possible force on 1000 meters of this wire?

(d) A square coil is constructed with 250 turns of wire. Each side is 3.75 cm. Calculate the torque on this

coil which has a current of 1.24 A, a magnetic field of 575 gauss, and the direction of the field is in the

plane of the loop.

(e) If each resistor has a value of R, calculate the effective resistance

between a and b.

6. (a) An electron is accelerated from rest through a potential of 500 volts. Calculate the radius of its circular

path in the earth's magnetic field (assumed to be exactly 1.00 gauss).

(b) Calculate the cyclotron frequency, in Hz, for a proton in a magentic field of 0.525 T.

(c) For the circuit shown calculate the current in the resistor 2.10 time constants after the switch is closed.

g = 250 V; R = 350 S; C = 1.75 :F

(d) If a bolt of lightning has a current of 1.2 × 10 A and a radius of 3.75 cm, what is the average current5

density?

(e) If copper has a density of 8.97 × 10 kg/m find the magnetic field needed to balance the weight of a3 3

copper rod whose diameter is 3.25 cm. The rod is carrying a current of 10,000 A and is horizontal.

The magnetic field is horizontal and perpendicular to the rod.

7. (a) Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 1.75 T.

(b) A galvanometer is built with a plane circular coil of radius 3.75 cm and 750 turns of wire. If the

magnetic field is in the plane of the coil, calculate the torque (in N@m) for a current of 175 milliamperes

and a field of 0.450 T.

(c) Determine the drift velocity for electrons in a round copper wire of radius 0.75 mm. The current is

15.0 A, the wire is 6.00 m long, the density of copper is 8.50 grams/cc. The atomic mass of copper is

65.0.

(d) A 12.0 volt battery has an internal resistance of 1.50 S. What is the power it can deliver to a load of

1.00 S?

(e) If the earth's magnetic field is 1.00 × 10 T, calculate the force on 10.0 m of wire which is-4

perpendicular to the field and carries a current of 1,750 A.

8. Electrons are run through a velocity selector as discussed in class with the following values of the fields: E =

125,000 V/m and B = 0.350 T.

(a) Calculate the velocity of electrons selected by this system;

(b) Protons are incident on the same system. Find the radius of the orbit of these protons after they leave

the velocity selector and pass through a region with B = 0.350 T perpendicular to the velocity.

9. (a) Calculate the time constant for a circuit containing a 11.0 pF capacitor and a 22,500

ohm resistor in series.

(b) Electrons from the sun arrive at earth with a velocity of 1.00 × 10 m/s. What is the7

radius of their orbit in a magnetic field of 1.00 gauss?

(c) What is the equivalent resistance between a and b if all resistors

have the value R.

(d) If the internal resistor of a 12.0 volt car battery is 0.01 ohm, calculate the power dissipated in the

internal resistance when the battery is connected to a load of 0.02 ohms.

(e) Calculate the drift velocity for electrons in a copper wire whose diameter is 1.20 mm if the wire is

carrying a current of 7.75 A. [n = 8.47 × 10 e /m ] 28 - 3

10. (a) Find the radius of the orbit of a proton with speed = 3.86 × 10 m/s in a perpendicular magnetic field of5

3650 gauss.

(b) Calculate the cyclotron frequency for electrons in a magnetic field of 250 gauss.

(c) Calculate the time constant for a single loop circuit containing a 150 V battery, a 75000 ohm resistor

and a 3.25 × 10 F capacitor. -12

(d) Determine the drift velocity in a semiconductor with 2.79 × 10 charge carriers/cm . The current is18 3

4.75 A in a circular wire with a diameter of 0.132 mm.

(e) All resistors have the same values, R. Calculate the effective

resistance between (a) and (b).

11 (a) Calculate the cyclotron frequency for electron in a magnetic field

of 367 gauss (in Hz).

(b) Calculate the drift velocity in a semiconductor with a carrier density of 3.46 × 10 carriers/cm . The17 3

current is 3.27 A in a wire with circular cross section and diameter 2.75 mm.

(c) All resistors have the same values, R = 100 S.

Calculate the effective resistance between (a) and (b).

Numerical answer.

(d) A 12.3 volt battery has an internal resistance of 0.75 ohm. What power will it deliver to an external

load of 1.25 ohms?

(e) Calculate the time constant for charging the capacitor if the resistor R =

17.500 S and the capacitance is 6.25 :F.

12. A circular coil of wire with 175 turns and a radius of 1.75 cm is suspended in a horizontal magnetic field. The

magnetic field is in the positive x-direction. The current in the coil is 0.750 A.

(a) Calculate the magnitude of the torque on the coil.

(b) Find the magnitude of the magnetic moment of the coil.

(c) In the top view shown, the torque is observed to move the coil in a counterclockwise direction. What

is the direction of ? Express this as an angle measured counterclockwise from the positive x-

direction.

13. A and B are two very long, straight wires. A carries a current of 27.0 A into the paper, and B

carries a current of 33.0 A out of the paper. If a = 17.0 cm, calculate the magnetic field, magnitude

and direction, at point P. Show clearly with a drawing how you define

the direction.

14. Given an equilateral triangle of wire of side a carrying a current I. Calculate

the magnetic field (magnitude and direction) at the center of the triangle, point

P.

15. A long copper pipe with thick walls has an inner radius a = 0.75 cm and an outer

radius b = 2.24 cm. It carries a current, uniformly distributed, of 9500 A, into the

paper.

(a) Calculate the magnitude of the magnetic field at R = 3.00 cm.

(b) Calculate the magnitude of the magnetic field at R = 1.75 cm.

(c) On a drawing indicate the direction of the magnetic field in (b).

16. Two semi-infinite wires are in the same plane. The wires make an

angle of 45/ with each other, and they are joined by a curved

section of wire that is an arc of a circle of radius R. If the wires

carry a current I, find the magnitude of the magnetic field at the

center of the arc of the circle (point P).

17. Four long, straight wires are arranged in a square perpendicular to the paper as

shown. (+ means current out of the paper, - means current into the paper.) The

sides of the square have length a = 2.75 cm. Calculate the force per unit length

(magnitude and direction using the coordinates shown), on wire 4.

1 3I = +3.00 A I = +6.25 A

2 4I = -4.75 A I = +5.25 A

18. Three very long wires are arranged in the configuration shown. The two lower wires are fixed

in position and carry identical currents out of the paper. The upper wire has a mass density of

1.50 kg/m. It has the same current as the lower wires, but in the opposite direction. Calculate

the magnitude of the current I that will support the upper wire in the position shown.

19. Use the Biot-Savart law to calculate the magnetic field at point P (in x,

y, z notation), due to a current of 5.00 amperes in the direction shown by

the arrows. The round portion of the wire is circular (R = 6.00 cm), and

P is at the center. For ease in grading, label the infinite straight

segments (1) and (2) as shown in the figure.

20 (a) Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 0.275 T.

(b) A circular loop of wire has 35 turns and a diameter of 11.0 cm. If the current in the wire is 7.25 A,

calculate the magnitude of the magnetic field at the exact center of the loop.

(d) Three long, straight wires are in a plane. They are each a distance a

apart. The currents are in the direction of the arrows, and have the

1 2 3magnitudes, I = 4.00 A, I = 3.00 A and I = 1.00 A. Calculate the

magnitude of the magnetic field at point P, which is in the same plane

as the wires. Take a = 1.00 cm.

(e) Electrons are accelerated from rest by a potential difference of 475 V. Calculate the magnitude of their

velocity.

(f) Three charges are arranged as shown. If *Q* = 4.0 × 10 C and!12

a = 3.00 × 10 m, calculate the electric potential at point P,!4

including sign, using the usual choice for the zero of potential.

o21 A long, hollow, cylindrical copper pipe, with outer radius R = 2.75 cm and

iinner radius R = 0.50 cm, carries a current of I = 1.46 × 10 A. The current is4

uniformly distributed.

(a) Calculate the magnitude of the magnetic field at r = 0.45 cm from the

center of the cylinder.

(b) Calculate the magnitude of the magnetic field at r = 3.75 cm.

(c) Calculate the magnitude of the magnetic field at r = 1.25 cm.

22 (a) Find the magnetic field at the center of a circular loop consisting of 25 turns of wire, with each turn

carrying 13.2 Amps. The radius of the loop is 0.25 m.

(b) Find the magnetic field in the interior of an ideal solenoid consisting of 1750 turns carrying a current of

0.375 Amps. The solenoid is 10.0 cm long and has an inside diameter of 1.2 cm.

(c) Find the cyclotron frequency of an electron in a magnetic field of 0.52 gauss.

(d) A cable carries a current of 1000 Amperes. Find the force on 250 m of the cable if there is a magnetic

field of 0.50 gauss (about the Earth's field) at right angles to the cable.

(e) Find the magnetic field at a distance of 10 cm from a bolt of lightning carrying a current of 30,000 A.

(A typical peak current.)

23. (a) Find the magnetic field 15 m from a long straight wire carrying a current of 125 A.

(b) Given a circular coil of radius 15 cm and 17 turns. If it carries a current of 1.25 A, find its magnetic

dipole moment.

(c) Two long parallel wires carry a current of 18 A. If they are 3 cm apart, calculate the force per meter on

each wire.

(d) In the circuit shown, calculate the time constant.

g = 10 V

C = 1000 pf

1R = 10 S4

2R = 2 × 10 S4

24 The drawing shows the cross section of three very long straight wires. They

are arranged in the form of an equilateral triangle of sides a. + means a

current out of the paper and - is a current into the paper. Calculate the

magnetic field, both magnitude and direction at point P. Show on a clear

drawing how you define any angle used to indicate the direction of the field.

1I = + 5.00 A

2I = - 3.25 A

3I = - 2.75 A

a = 1.25 cm

25 The diagram shows a cross section of long, straight wires perpendicular to the

paper. + indicates currents out of the paper. - indicates currents into the paper.

Find the magnitude and direction of the magnetic field at point P. The four wires

form a square. P is at the center of the square.

26. Given three long straight wires, A, B and C that are perpendicular to the

plane of the paper. Calculate the magnetic field, magnitude and direction, at P. Use the

x-y axes shown. Positive (+) current means current coming out of the paper, negative (-) is

A B Ccurrent going into the paper. I = +8.00 A; I = - 4.00 A; I = - 11.00 A

o27 Consider modeling a bolt of lightning as a long cylinder with a current density given by j = j (1 - "R ), where3

o o" is a constant and j = 0 at the outside radius R where R = 4.50 cm. The total current is 65,500 Amperes (a

typical number).

(a) Calculate " (numerical value with units).

o(b) Find j (numerical value with units).

(c) What is the magnetic field a distance R = 2.50 cm from the center of the current distribution? [If you

cannot do (a) and (b) do this symbolically.]

28. In the diagram + means currents out of the paper, ! means currents into

the paper. There are two long straight wires, A and B perpendicular to the paper.

A, B and P are at 3 corners of a square.

(a) Calculate the magnetic field, magnitude and direction, at point P.

Measure angles counterclockwise from the positive x-direction.

(b) A third wire is placed at P perpendicular to the paper. Determine the

force per meter, magnitude and direction, on the third wire if its current is

!4.75 A.

29. Three long, straight wires are perpendicular to the plane of the paper. Wire A has

current into the paper. Wires B and C have current out of the paper. Each wire has a

current of magnitude 1.50 A. [a = 5.65 cm]

(a) Find the magnetic field, magnitude and direction, at point P which is in the

center of the square of which A, B and C are corners.

(b) Calculate the force per unit length, magnitude and direction, on a fourth wire at

Q, the remaining corner of the square. The current in Q is 1.50 A out of the

paper. The wire Q is parallel to A, B and C.

o30. A long, straight, copper pipe has an inside radius a and an outside radius b. The pipe carries a current I of

4.75 × 10 A out of the paper. [a = 1.75 cm; b = 2.15 cm]3

(a) Calculate the magnitude of the magnetic field a distance 1.60 cm from the

center of the pipe.

(b) Calculate the magnitude of the magnetic field a distance 2.25 cm from the

center of the pipe.

(c) Calculate the magnitude of the magnetic field a distance 2.05 cm from the

center of the pipe.

(d) In (c), is the magnetic field direction clockwise or counter-clockwise?

31. (a) Calculate the cyclotron frequency for an electron in a magnetic field of 1.22 T.

(b) Take the direction of the earth's magnetic field as being exactly N-S. If its value is 0.550 gauss, what is

the force, magnitude and direction, on one meter of wire carrying 150 amperes from west to east (the

direction of the conventional current).

(c) Calculate the radius of the orbit of a proton in a magnetic field of 0.300 T, if the velocity of the proton

is v = 2.4 × 10 m/s. The plane of the orbit is perpendicular to the magnetic field. 4

(d) Calculate the time constant for a simple RC circuit if R = 450 kS and C = 670 pF.

(e) Calculate the magnetic field, in Tesla, needed for a velocity selector for electrons if the velocity desired

is 2.00 × 10 m/s and the electric field is 4.00 × 10 V/m. 3 4

132. For the circuit shown the capacitor is initially uncharged. Initially S is closed. At

2 1t = 0 S is closed and S is still closed.

(a) Calculate the charge on the capacitor at t = 4,

(b) Find the voltage across the capacitor when t = 1.50 J.

1(c) Calculate the current in R at t = 0.

1(d) With the capacitor fully charged S is opened. Calculate the time constant

for discharging C.

1 2(e) With S and S closed, calculate (numerical value) the time constant for

charging C. (No short cuts from advanced courses allowed.)

1 2 3g = 100 V; R = 150 kS; R = 175 kS; R = 125 kS; C = 800 pF

33. (a) Calculate the time constant for charging the capacitor in the circuit

shown.

(b) Calculate the magnitude of the magnetic field, in Tesla, at the exact

center of a circular loop of wire carrying a current of 14.5 A. The diameter of the loop is 17.0 cm.

(c) Assume the earth's magnetic field is exactly 1.00 gauss. Calculate the radius of the orbit of an electron

moving perpendicular to this field at a velocity of 2.1 × 10 m/s. 6

(d) Two parallel wires carry a DC current of 55.0 amperes. If the wires are 4.00 m apart, calculate the

magnitude of the force on each meter of wire.

(e) Calculate the capacitance of a parallel plate capacitor if the plates are circular with diameter 17.0 cm.

The plates are 0.25 mm apart (in air).

34. A velocity selector is created with the fields E = 1200 V/m and B = 0.300 T.

(a) For protons incident on this selector, what velocity is passed?

(b) What is the cyclotron frequency for protons in the 0.300 T magnetic field?

(c) What is the radius of the cyclotron orbit for protons passed by the velocity selector in (a) in the

magnetic field of 0.300 T?

35. Three long parallel wires are arranged perpendicular to the paper at three corners of a

square as shown. Positive (+) currents are out of the paper. Calculate the magnetic field,

magnitude and direction, at point P, the fourth corner of the square.

1 2 3I = + 3.30 A; I = + 4.75 A; I = - 6.00 A; a = 7.00 cm

36. (a) Calculate the cyclotron frequency for an electron in a magnetic field of 0.81 Tesla.

(b) Calculate the time constant for a simple series RC circuit where R = 600 S and C = 325 pF.

(c) Calculate the magnetic field in the exact center of a circular loop of wire carrying a current of 3.75 A.

The diameter of the loop is 6.75 cm.

(d) If the observed velocity of an electron is 3 × 10 m/s and the electric field is 5.00 × 10 volts/meter,3 4

find the magnetic field, in Tesla, needed for a velocity selector.

(e) Two long parallel wires each carry a current of 75 A (DC current). If the wires are 0.75 m apart,

calculate the magnitude of the force on each meter of wire.

37. In the drawing, A, B and P are at corners of a square of side a. + currents are out

of the paper; - currents are into the paper. A and B are long wires perpendicular to

the paper. Calculate the magnetic field, magnitude and direction, at point P. Use

the coordinate system shown.

A Ba = 2.50 cm; I = -7.00 A; I = +5.00 A

38. Shown is the circular cross section of a long cooper rod of diameter 1.00 cm. The rod

carries a current of 75 A into the paper. Assume the current is uniformly distributed.

(a) Calculate the magnitude of the magnetic field at point A, a distance a = 2.00 cm

from the center of the rod.

(b) Calculate the magnitude of the magnetic field at point B, where b = 0.25 cm from

the center of the rod.

39. Three very long wires are arranged in the configuration shown. The two lower wires are fixed in

position and carry identical currents out of the paper. The upper wire has a mass density of

2.50 kg/m. It has the same current as the lower wires, but in the opposite direction. Calculate the

magnitude of the current I that will support the upper wire in the position shown. u means out of

and q means into the paper. (a = 10.0 cm)

40. Three long wires, Î, Ï, and Ð, are perpendicular to the paper. They carry currents with

magnitude given with directions indicated. u is out and q is into the paper. Calculate the

magnetic field, magnitude and direction, at point P. Use the x-y coordinates shown. State

clearly the direction.

1 2 3I = 10.00 A; I = 5.00 A; I = 8.00 A; a = 10.0 cm

41. (a) Take the earth’s magnetic field as 1.27 × 10 T at 80° from the horizontal in Salt!4

Lake City. Calculate the magnetic flux through 6.25 m of the physics parking lot. __________2

(b) Given a circular loop of wire of R = 2.25 cm. There is a magnetic field

perpendicular to the paper which can be described by B(t) = (945 + 1.75 × 10 t)!4

Tesla. Calculate the magnitude of the emf that appears in the loop.

(c) Given a circular loop of wire. There is a magnetic field into the paper and its

magnitude is decreasing. What is the direction of the current in the loop–clockwise

or counterclockwise?

(d) Calculate the cyclotron frequency, in Hz, for an electron in a magnetic field of 0.333 T.

(e) Calculate the magnetic field at the surface of a large wire (diameter = 6.25 cm) carrying a current of

4750 A.

42. At (A) a velocity selector is operated with an electric field of 1.62 × 10 V/m in the5

direction shown.

(a) Calculate the magnitude of the magnetic field necessary to select the velocity v =

1.27 × 10 m/s for singly positively charged particles moving to the right.5

(b) What is the direction of this magnetic field–into or out of–the paper? Explain.

(c) The same magnetic field exists in region (B). There is no electric field in region

(B). Calculate the value of R for protons.

(d) Calculate the displacement, in meters, at point (C) between nuclei of U-235 and

U-238 through the same velocity selector. Take the mass numbers as exact,

which they are not in nature.

43. Three long, straight wires are perpendicular to the paper at points Î, Ïand Ð. Positive (+) currents are out of the paper, negative (!) currents

into the paper. Calculate the magnetic field, magnitude and direction, at

point P. The direction should be specified as an angle measured from the

positive x-axis. Point P is directly above Ð.

1 2 3I = +3.27 A; I = !1.75 A; I = !1.10 A

44. (a) Show in detail how to calculate the magnetic field at point P using the Biot-

Savart law and evaluate the answer. Consider the wire to be infinite at both

ends..

(b) If the magnitude of the current in the wire is 5.50 A, calculate the magnitude

of the field at P, if a = 2.63 cm

(c) What is the direction of the magnetic field at P (into or out of the paper)?

45. (a) An ideal solenoid with circular cross section has 4750 turns in a length of 10.2 cm. The radius of its

cross section is 0.350 cm. When the current is 11.2 A, what is the outward force per meter on th wire?

(b) A coil with 3 turns and a diameter of 2.75 cm is rotated about the vertical axis in a

horizontal magnetic field of 850 gauss. The frequency of rotation is 300 Hz. Write a

complete expression, with all quantities numerically evaluated for the voltage at A, in the

form g = D cos Tt.

(c) Calculate the magnetic field at the surface of a big fat wire carrying 17,500 A. The wire has a circular

cross section with a diameter of 3.50 cm.

(d) Assume that you tried to cancel the earth’s magnetic field with a coil wrapped around the equator. If

the coil had 1000 turns, what current would be needed to created a magnetic field of 1.00 gauss at its

center?

(e) If bismuth has 4.7 × 10 charge carrier/cm and these carriers all have the electron charge (not, in fact,17 3

true), calculate the Hall voltage for a long strip of Bi 1.00 cm wide and 0.100 cm thick, in a magnetic

field perpendicular to the wide face, of 3.25 T if the current is 0.115 A.

1 246. In the drawing, y is the vertical direction. The currents I and I are in the

1 2direction given. The wire carrying I is fixed, the wire carrying I can move up

2and down in the y direction only. The wire carrying I has a mass density of

0.035 g/meter.

1 2 2(a) If I = 10.0 A, what must the current I be to suspend I at

a = 1.25 cm?

1 2 2(b) If I = I , what must the value be to suspend I at a = 1.25 cm?

47. Three long straight wires are carrying current perpendicular to the paper.

r means current out of the page, s means current into the page. Using the

current values given, calculate the magnetic field at point P.

(a) Express the magnetic field in î, ¯ notation.

(b) Calculate the magnitude and direction of the field. Express the

direction as an angle measured counter clockwise from the positive

x-direction. Show on a drawing how you define this angle.

1 2 3a = 4.50 cm; I = +10.25 A; I = -5.75 A; I = - 2.30 A

48. In the drawing, the horizontal lines represent conducting rails.

The rod, A, has a mass of 75.0 g. The switch A is closed at t = 0.

There is a magnetic field perpendicular to the paper, and INTO

the paper, whose strength is B = 0.930 T.

(a) Which direction, right or left, would the rod move?

(b) Just after t = 0, what is the force on the rod?

(c) Just after t = 0, what is the acceleration of the rod?

(d) If the rod moves without friction, what is its maximum

speed?

(e) What is the time constant for the approach to this

maximum speed?

g = 75.0 V; R = 2.00 S; R = 9.50 cm

49. (a) Calculate the magnetic field (in Tesla) 0.015 m from the center of long, straight wire carrying a current

of 16.2A.

(b) Calculate the force due to the earth’s magnetic field on 150 m of electric power line when the current is

1670 A if the earth’s is 0.75 gauss and is perpendicular to the power line.

(c) Find the radius of the path of a proton (m = 1.67 × 10 kg) in a magnetic field of 0.11 gauss, if its !27

velocity perpendicular to the field is 4.20 × 10 m/s. ____________5

(d) If R = 375 ohms, calculate the power dissipated by the top resistor if a battery of

155 volts is applied between a and b.

(e) Calculate the charge on the capacitor 1.50 time constants after the switch is closed.

g = 175 V; R = 275 S; C = 15 pF

50. Three long straight wires are perpendicular to the paper at the points labeled A, B, C.

The currents are given below. + is out of the paper, ! is into the paper.

(a) Calculate the x-component of the magnetic field at P (with sign).

(b) Calculate the y-component of the magnetic field at P (with sign).

(c) Calculate the magnitude of the resulting magnetic field at P.

(d) Calculate the direction of the magnetic field at P, as an angle measured co

unter clockwise from the positive x-axis.

A B CI = +15.2 A; I = +12.7 A; I = !17.2 A; a = 3.65 cm

51. A velocity selector is set up using an electric field of 10,000 V/m.

(a) Calculate the magnetic field (in Tesla) necessary to select a velocity of 7.50 × 10 m/s.4

(b) Calculate the difference in the radii of two atoms at this velocity in a magnetic field

perpendicular to their velocity of 0.912 T. The two atoms are and . Each has

one electron missing. (1 amu = 1.66 × 10 kg)!27

o o52. A long, straight wire carries a current given by I = I e , where I and k are-kt

constant. Nearby is a rectangular loop of wire with the dimensions shown.

Both the straight wire and the rectangular loop are in the plane of the paper.

(a) Calculate an expression for the flux through the rectangular loop as a

function of time.

53. Given a long, straight solenoid of inside diameter 3.75 cm, length 55.0

cm, consisting of 8700 turns of very fine wire. Inside the solenoid is a

coil of 7 turns of wire, whose plane is oriented as shown. The radius of

the inner coil is 0.15 cm.

(a) If the current in the solenoid is 1.75 A, calculate the flux through each

turn of the inner coil.

54. A long cylindrical conductor of radius R carries current I of uniform density

J = I/BR . Find the magnetic flux per unit length through the area indicated in2

the drawing. This area is a long plane. One side is the center of the cylinder, the

other is the outside radius.

556. Two long, straight wires are parallel and a distance d apart.

(a) Calculate the magnitude of the magnetic flux through the rectangular region

oshown. Each wire carries a current i , in the direction shown by the arrows.

56 Consider a toroid with a rectangular cross section and dimensions shown. The

toroid has 975 turns of wire and carries a current of 2.75 A.

(a) Calculate the magnetic field within the toroid at any value of r between r = a

and r = b. (This requires a numerical answer, except for r.)

(b) Calculate the flux crossing a section of the toroid of width c, between r = a

and r = b/2. (This requires a numerical answer.)

a = 2.50 cm b = 12.50 cm c = 4.00 cm

57 In the drawing shown the wire carrying current and the rectangular loop are both in the plane

oof the paper. The wire is long. The current in the wire is given by I = I cos Tt. The positive

direction is shown.

(a) Determine the magnetic flux through the rectangular loop in terms of the current in the

wire and the geometry.

(b) Find the current in the resistor (magnitude) as a function of time.

(c) Clearly explain which direction (clockwise or counter-clockwise) the current in the

rectangular loop is going at t = 0.

58 (a) Calculate the radius of the path of an electron moving at 7.50 × 10 m/s perpendicular to a magnetic4

field of 0.35 T.

(b) Determine the magnitude of the force on a horizontal electric power line 200 meters long carrying a

current of 1,750 A. Assume the earth's magnetic field is vertical and has a value of 0.450 gauss.

(c) For the circular loop shown the magnetic field is perpendicular to and into the paper. If the

loop is a conductor and the magnetic field is decreasing, is the induced current clockwise or

counterclockwise?

o o(d) A magnetic field in a region of space has B = B sin Tt, with T = 266 rad/s and B = 0.0150

T. Find the EMF that appears in a coil of 7 turns whose plane is perpendicular to this field.

The coil has a diameter of 27.5 cm.

(e) If the earth's magnetic field is at an angle of 70° from the horizontal as shown, what is the

magnetic flux through an area 7.00 × 2.70 m in the Physics parking lot. Take the earth's field as

0.800 gauss and assume the parking lot is level.

59. A long straight wire carries a current I in the direction shown. Calculate the magnetic flux through the

rectangular area shown. The long sides of the rectangle are parallel to

the wire, the short sides are perpendicular to the wire.

60. (a) Electrons are moving with a velocity of 3.30 × 10 m/s perpendicular to a magnetic field of 0.119 T. 6

Calculate the magnitude of the electric field perpendicular to their path such that they move in a

straight line.

(b) Show is a top view of a circular coil of 12 turns and radius 3.27 cm. If the

omagnetic field can be represented as B = B cos Tt, calculate an expression

for the EMF induced on the coil.

(c) The drawing shows a rectangular cross section of a toroid of 375 turns with a

current of 1.75 A. The inner radius of the toroid is 12.65 cm. Calculate the flux

crossing the hatched area. a = 1.15 cm; b = 2.35 cm

(d) If the earth's magnetic field is 1.10 × 10 T and is directed at an angle of 67.0° from!4

the horizontal, calculate the flux through an area 10.0 m × 15.0 m of the physics

parking lot.

(e) A wire carries a current from west to east horizontal to the ground. What is the direction of the force

on this wire due to the earth's magnetic field (assumed to be in a north-south direction).

61. (a) Calculate the magnetic flux (in T@m ) through a section of the physics parking lot that is 5 m × 10 m. 2

Take the earth’s field as 0.75 gauss, and its direction as 70° from the horizontal.

(b) Find the magnitude of the magnetic force per unit length on a power line if the current is 150,000 A

and the earth’s magnetic field is 0.75 gauss and is perpendicular to the wire.

(c) Calculate the cyclotron frequency, in Hertz, for electrons in a magnetic field of 0.75 gauss.

(d) A circular loop of wire has 11 turns, carries a current of 1.50 A, and has a diameter of 7.50 cm.

Calculate the magnetic field at the exact center of the loop.

(e) For the circular loop shown the magnetic field is perpendicular to and out of the paper. If the

loop is a conductor and the magnetic field is increasing, is the induced current clockwise or

counterclockwise? (B = u means out of the paper.)

62. The long vertical wire carries an upward current given by

oI = I e!kt

owhere I is the current at t = 0. The conducting loop is in the plane of the paper.

(a) Calculate the magnetic flux through the rectangular loop at t = 0.

(b) Find the current in the rectangular loop as a function of time.

(c) Explain clearly why this current in the rectangular loop will be either clockwise

(CW) or counter clockwise (CCW).

63. In the drawing, the horizontal lines represent conducting rails. The rod, A, has a mass of 75.0 g. The switch

A is closed at t = 0. There is a magnetic field perpendicular to the paper, and INTO the paper, whose strength

is B = 0.930 T.

(a) Which direction, right or left, would the rod move?

(b) Just after t = 0, what is the force on the rod?

(c) Just after t = 0, what is the acceleration of the rod?

(d) If the rod moves without friction, what is its maximum speed?

(e) What is the time constant for the approach to this maximum

speed?

g = 75.0 V; R = 2.00 S; R = 9.50 cm

64. (a) Given a rectangular toroid with the cross section shown and N turns of

wire, calculate its inductance.

(b) If N = 360, a = 2.75 cm, b = 7.25 cm and c = 1.25 cm, calculate the

numerical value for the inductance.

(c) When the current is 1.65 A, calculate the magnetic energy stored

between R = 1.25 cm and R = 4.85 cm.

65. Given a long, straight copper wire of circular cross section. The radius of the wire is 1.50 cm. (It is a big

wire.) The wire carries a current, uniformly distributed across the cross section, of 12500 A.

(a) Find an expression for the magnetic field at an arbitrary point inside the wire.

(b) Find an expression for the magnetic energy density at an arbitrary point within the wire.

(c) Find the magnetic energy stored within a 10 m length of the wire.

66. A long straight wire carrying a current is in the same plane as a rectangular loop

of wire with the dimensions shown. The wire has a resistance, R, as shown.

o(a) If the current in the wire is given by I = I sin Tt, calculate the current

through R as a function of time. Assume the positive direction of the

current in the wire is as shown by the arrow, and that the positive current

in the rectangle is clockwise.

(b) Now the current in the long wire has a constant value I (new situation),

and the shape of the rectangle is changed by changing the value of c at a

steady rate of dc/dt = A m/s. Calculate the current in the resistor,

including its sign, using the convention in (a).

67. A large (R = 3.65 m) circular loop of wire (1 turn) carries a current given by I = 35.0 cos

400 t Amperes. A very small circular coil of 15 turns and radius r = 2.25 mm is at the

exact center of the loop. The time is indicated by t.

(a) If the plane of the small coil makes an angle of 15.0/ with the plane of the loop,

calculate the magnitude of the emf induced in the small coil as a function of time.

(b) If the current in the loop is fixed at 45.0 A and the small coil rotates about the axis

A at an angular speed of 275 rad/s, calculate the magnitude of the emf generated

in the small coil as a function of time. The diameter of the small coil is parallel to

A.

68. Given a long, straight wire carrying current to the right as shown. A

rectangular loop of wire is placed near the wire as shown. If the current

oin the wire is given by I = I sin Tt, find an expression for the voltage

across the resistor R as a function of time.

69. A conducting rod of mass M and length L slides without friction on two

long, parallel, horizontal rails. A uniform magnetic field B fills the

region in which the rod is free to move. B is out of the paper. A battery

is attached to the rails and supplies a constant emf through a resistor R.

The rails and rod have negligible resistance.

(a) Find the direction of the current (clockwise or counterclockwise)

such that the rod moves to the left.

(b) When the rod moves with velocity v, what is the magnitude of the emf generated in the rod?

(c) Find a differential equation for the speed of the rod as a function of time. (Hint: You have seen this

equation before.)

(d) Find an expression for the speed of the rod as a function of time.

70. A conducting, resistanceless rod slides down a pair of resistanceless

frictionless rails inclined at an angle of 25.0/ above the horizontal. The

entire region is in a uniform B field of 0.365 T that is in the vertical

direction. The resistance of the resistor is R = 10.0 S. The distance

between the rails is R = 0.650 m. The mass of the rod is 8.00 grams.

(a) Find the steady state (or terminal) velocity of the rod as it slides

down the rails.

(b) Find the time constant for the approach to steady state motion.

71. A conducting rod of mass 1.75 kg slides without friction on two horizontal

conducting rails that are 10.0 cm apart. It starts from rest at t = 0. At t = 0 a

steady current I = 25.0 A, is turned on. The rod travels 2.09 m to the right in

the first 3.00 seconds.

(a) Find the magnitude and direction (up or down) of the magnetic field

(assumed uniform and vertical).

(b) Calculate the emf generated in the rod as a function of time.

o72. A very long straight wire carrying a constant current I is in the same plane as two

conducting rails. A rod of resistance R and mass M can slide without friction on

the rails. The rod experiences a constant external force F in the direction shown.

The rod remains perpendicular to the rails.

(a) Calculate the emf generated in the rod as a function of its velocity.

o(b) Calculate the acceleration of the rod (in terms of I , R, a, x, v, M as needed).

(c) Calculate the time constant for the approach of the motion to a steady state.

73. A conducting rod moves on two

parallel frictionless conducting rails. The only resistance is that

shown as R. There is a battery in series with R. The rod has a

mass m of 1.35 × 10 kg. The magnetic field is in the vertical-2

direction, and has the magnitude of B = 4.75 T. g and R have the

values shown. The rod starts at rest (t = 0).

(a) At t = 0, what is the acceleration of the rod? Take the

direction up the incline as positive.

(b) Assume the rails are infinitely long in both directions and

B is the same everywhere. What is the velocity of the rod

at very long times?

(c) What is the time constant for the approach of this system

to the steady state?

o74. A bolt of lightning bolt is modeled as I = I sin At between t = 0 and

t = B/A. Calculate the EMF that appears around the cross hatched loop

of metal fence with the dimensions shown, when the lightning strikes

the metal post.

75. (a) Calculate the magnetic field at the center of a circular coil of 45 turns carrying a current of 1.75 A.

The coil has a diameter of 35.0 cm.

(b) Calculate the cyclotron frequency (in Hz) for electrons in a magnetic field of 4750 gauss.

o o(c) Imagine that the magnetic field in some region of space varies as B = B + at + bt , where B , a and b2

are positive constants. Calculate the maximum emf that is induced at t = 10 s in a coil of 45 turns and a

diameter of 35.0 cm if placed in this region of space.

(d) If the capacitor shown has a charge of 45.0 × 10 C at t = 0, calculate the!3

voltage across the capacitor at t = 5.00 s. © = 4500 :F, R = 4800 S).

SIDE VIEW

(cross section)

TOP VIEW

(e) A wire carries a current in the direction shown. The wire and the loop are in the

plane of the paper. If the magnitude of the current in the wire is decreasing with

time, is the induced current in the loop clockwise or counterclockwise?

(f) A bolt of lightning with a peak current of 65,000 A strikes a tall metal post stuck in the ground.

Calculate the peak value of the magnetic field 10.0 cm from the post.

76. The toroidal coil shown has a rectangular cross section and is wound with 953 turns of wire. The distances a

and b are measured from the center of the doughnut.

(a) Calculate the value of the magnetic field inside the coil for any r between a and b (a < r < b) and for

any value of the current I.

o(b) If I = I e where k is a constant, calculate the current at any time t in the single rectangular loop of!kt

wire inside the toroid. The loop has dimension a/2 and c/3 as shown, and its left edge, as in the picture,

is at r = a.

77. A coil, which has 15 turns, is in a region of space where the magnetic field is given by

o oB = B cos Tt, where B = 1.50 × 10 T and T = 266 rad/s. The coil has a diameter of!5

5.25 cm. The wires from the ends of the coil are close together and outside the

magnetic field there is a resistor R = 150 S between the coils ends. If the coil is in the

paper, B is perpendicular to the paper. Calculate all possible numbers.

(a) Find the magnitude of the EMF generated in the coil as a function of time.

(b) Calculate the magnitude of the current that flows through the resistor as a

function of time.

(c) Determine the power dissipated in the resistor as a function of time.

78. In the drawing shown, everything is in the plane of the paper. The positive direction for

current is given by the arrow. The rectangle is

made of a conductor.

(a) For a current I, calculate the magnitude

of the flux through the rectangle.

o(b) If the current is given by I = I sin Tt,

calculate the magnitude of the voltage as

a function of time across R.

(c) For the current given in (b) explain

clearly the direction of the induced

current (clockwise or counterclockwise)

in the rectangle at time t = 0.

Side View

79. A conducting rod moves on two parallel conducting rails that are a distance w apart. The only resistance is the

R shown. The magnetic field is perpendicular to the paper, and

into the paper, and is everywhere in the drawing.

(a) If the rod is moved to the right with constant velocity v,

calculate the current through R.

(b) If the rod is moved to the right with constant force F,

calculate the velocity of the rod at a very long time. (the

rails are infinitely long.)

80. A rectangular conducting wire has the shape and dimensions

as shown. The upper wire and the rectangle are in the plane

of the paper.

o(a) If the current in the upper wire has a steady value I ,

calculate the magnetic flux through the rectangular

loop.

o(b) If the current in the upper wire is given by I = I sin Tt,

calculate the current in the rectangular loop as a function of time.

(c) If the positive direction of the upper current is to the right, as shown, calculate the direction of the

current in (b) at t = 0, clockwise or counterclockwise. For full credit, state clearly your reasoning.

81. A conducting rod of mass M and length L slides without friction on two long, parallel rails. As shown in the

side view, the rails are at 15° from the horizontal. A uniform vertical magnetic field B fills the region in

which the rod is free to move. B is out of the paper in the top view. A battery is attached to the rails and

supplies a constant emf through a resistor R. The rails and rod have negligible resistance.

(a) Find the direction of the current from the battery (clockwise or counterclockwise) such that the rod

moves to the right.

(b) When the rod moves with velocity v, what is the magnitude of the induced emf generated in the rod?

(c) What is the minimum voltage for g (battery) such that the rod will move to the right?

(d) Find the differential equation for the speed of the rod as a function of time.

(e) Find an expression for the speed of the rod as a function of time. Express the result in terms of g, R, M, L,

B, and angles, as needed.

82. Two long conducting rails are in the horizontal plane. There is a

uniform magnetic field, B, perpendicular to the paper and out of the

page. Starting at rest, a constant force F is applied to a conducting rod

of mass M, as shown. The rails are as long to the right as needed.

(a) Find the direction, clockwise or counter clockwise, for the

induced current.

(b) Determine the final velocity of the rod.

(c) Obtain a formula for the velocity of the rod as a function of time.

Show all details.

Top View

83. In the drawing the horizontal lines represent conducting rails that

are very long. The conducting rod, mass m, slides (no rolling)

without friction along the rails. It is acted on by a constant

horizontal force, F.

(a) When the speed of the rod is v, calculate the current in the

resistor.

(b) Obtain an expression for the limiting speed at which the

rod will move.

(c) Obtain an expression for the time constant for the approach

to the limiting speed.

84. (a) Calculate the speed of electrons that move in a straight line through a region that has a vertical E-field of

1750 v/m and a perpendicular magnetic field of 1.35 T. ___________________________________

(b) Protons are moving perpendicular to a magnetic field of 1.32 T. The radius of the circle they move in is

found to be 1.75 cm. Calculate their velocity. ___________________________________________

(c) An ideal solenoid has 6750 turns of wire, and a total length of 0.725 m. Calculate the magnetic field at its

center. __________________________________________________________________________

(d) A DC electric power line (they do exist) is carrying a current of 10,000 A. Calculate the force on 20.0 m

of this wire due to the earth’s magnetic field. Assume the component of the earth’s field perpendicular to

the wire is 0.750 gauss. _____________________________________________________________

(e) Assume that a wire carrying a current of 1.00 A is wrapped around the earth’s equator. How many turns

would be needed to generate a field of 10.0 gauss at the earth’s center? Take the radius of the earth as

6.38 × 10 km. ____________________________________________________________________3

85. The drawing shows three long straight wires perpendicular

to the paper. Each carries the same current, 1.75 A.

+ means out of the paper; ! means into the paper.

a = 1.65 cm

(a) Calculate the magnitude of the magnetic field at A

due to each of the three wires.

(b) Calculate the magnetic field vector at A due to all

three wires, magnitude and direction (not just

components).

86. The drawing shows three long straight wires perpendicular to the paper.

Wire Î has a current of 5.00 A out of the paper, wire Ï has an equal

current out of the paper. a = 2.75 cm

(a) If wire Ð has a current of 10.0 A, calculate the magnitude of the net force

per/meter on Ð.

(b) What must be the direction of the current in Ð be so that the net force is

upward? Explain clearly.

87. The drawing shows a long straight wire carrying current to the right.

o(a) If the current in the wire is I , calculate the magnetic flux

through the rectangle beside the wire. Do the

calculations; don’t just put down a formula.

(b) If the rectangle is a conductor with the resistance given,

calculate an expression for the current in the rectangle as

a function of time if the current in the straight wire is

ogiven by I(t) = I e .!kt

(c) What will the direction of the current (clockwise or

counter clockwise) in the rectangle be? Explain.

88. In the circuit shown, the switch is closed for a long time and then opened at t = 0.

(a) Calculate the current in the inductor at t = 1.32 × 10 s.-6

2(b) Calculate the voltage across R at t = 0.87 × 10 s.-6

2(c) Calculate the voltage across R at t = -0.100 × 10 s.-6

(d) Write three equations governing the behavior of the circuit with the switch

closed. Use the current labels given. Solve these to obtain the differential

equation governing the time rate of change of current in the inductor, and

from this calculate a numerical value for the time constant.

1 2 3g = 350 V, R = 250 S, R = 275 S, R = 125 S, L = 4.25 mH

1 2 389. In the figure shown, E = 100 volts, R = 10 ohms, R = 20 ohms, R = 30 ohms,

and L = 2 henry.

1 2(a) Find the value of i and i immediately after S is closed.

1 2(b) Find the value of i and i a long time later.

1 2(c) Find the value of i and i immediately after S is opened again.

1 2(d) Find the value of i and i a long time later.

90. In the drawing shown the network initially has no current. The switch S is

closed for two time constants. Then the switch is opened. Call this t = 0.

(a) Calculate the time constant for the circuit after the switch is opened.

3(b) Calculate the current in R , 1.80 × 10 seconds after t = 0.-6

4(c) Calculate the current in R at long times if the switch is closed.

1 2R = 1.26 mS = 1.26 × 10 S R = 23.0 kS = 2.30 × 10 S6 4

3 R = 50.0 kS = 5.00 × 10 S L = 13.0 mH4

g = 65.0 V

1 291. In the circuit shown g = 9.00 volts, R = 155 × 10 ohms, R = 195 × 10 ohms,3 3

and L = 3.00 mH. The switch S is closed at t = 0.

(a) Find the time constant for the circuit.

1(b) Find the value of the voltage across R when t = 2J.

2(c) Find an expression for the voltage across R as a function of time.

92. The switch S is closed at t = 0.

2(a) Find the current in R as a function of time.

(b) Write the differential equation for the current in L as a function of

time.

2(c) After the current in R has reached a steady state S is opened. Call

1this a new t = 0. Find the current in R as a function of time. Specify

its direction.

93. A toroidial inductor has 1742 turns and the physical dimensions shown in

cross section.

(a) Derive the formula for the inductance of a toroid and calculate its

value for this case.

(b) Calculate the energy stored between R = 5.00 cm and R = 6.00 cm

when the current is 0.275 A.

94. Given the circuit shown.

(a) Calculate the current through the inductance a long time after the

switch is closed. The inductance is assumed to have zero

resistance.

(b) Calculate the time constant for the approach to equilibrium after the

switch is closed. Show your work. You must show that you know

how to do this calculation from the beginning.

(c) After the switch is closed for a long time, it is opened at t = 0.

2Calculate the voltage across R at t = 6.25 × 10 s.-4

1 2 3 g = 125 V, R = 1000 S, R = 3000 S, R = 2000 S, L = 1.25 H

95. In the circuit shown the switch is closed at t = 0.

(a) Calculate the current in the inductance after 1.75 time constants have

elapsed.

(b) The switch is opened after being closed for 1.75 time constants in (a).

Calculate the time constant for the decay of the current in the

inductance.

(c) Calculate the time constant for the growth of the current in the

inductance with the switch closed. You must develop the differential

equation governing the current as function of time. For convenience in

grading use the branch current labels and directions given in the

drawing. No shortcuts.

1 2 3g = 150 V, R = 8,500 S, R = 15,700 S, R = 15,700 S, L = 3.25 mH

96. In the circuit shown, the switch is closed for a long time.

(a) Calculate the magnitude of the current in the inductance.

3(b) Calculate the magnitude of the current in R .

(c) If the switch is now opened at t = 0, calculate the magnitude of the

3current in R 2.50 time constants after t = 0.

(d) Calculate the time constant with the switch closed. Show ALL details,

as done in class.

1 2 3g = 125 V; R = 75 S; R = 100 S; R = 150 S; L = 13.0 mH

97. A long straight wire carries a current of 4.75 A.

(a) Find the magnitude of the magnetic field at a distance

r = 1.75 × 10 m from the wire.!2

(b) Calculate the magnetic energy stored in a region of space that is a tube

parallel to the wire of inner radius a = 3.20 cm and outer radius b =

7.70 cm with a length of 2.75 m.

98. In the circuit shown the switch has been open for a long time. It is closed at

t = 0.

3(a) Find the current in R at t = 0 (numerical value).

3(b) Find the current in R at t = 4 (numerical value).

3(c) Calculate the current in R after 1.70 time constants have elapsed after

t = 0 (numerical value).

(d) Set up a complete set of junction and loop equations and calculate the

time constant (numerical value) as show in class.

1 2 3g = 12.0 V; R = 600 S; R = 300 S; R = 450 S; L = 42.0 mH

99. A coaxial cable consists of a concentric inner wire of radius a and an outer conductor of inner

radius b. The same current flows into the paper on the inner conductor and out on the outer

conductor.

(a) Calculate the flux between the two conductors for a length R of this cable.

(b) Determine the inductance per unit length of this cable.

100. In the circuit shown, the switch is closed at t = 0 after being open for a long

time.

3(a) Calculate the current in R at t = 0

2(b) Calculate the current in R at t = 4.

(c) Calculate the current in the inductance 1.5 time constants

after t = 0.

(d) If the switch is opened 1.5 time constants after t = 0, write a complete

expression for the current in the inductor as a function of time with all

numerical quantities evaluated. Take the arrow as the positive

direction.

1 2 3g = 175 V; R = 100 S; R = 150 S; R = 250 S; L = 4.75 :H

101. For the circuit shown, the switch S is closed at t = 0 after being open for a

long time.

3(a) Just after t = 0, calculate the current in R .

3(b) At t = 4 calculate the current in R .

(c) At t = 4 the switch is opened. Calculate the magnitude of the

3current in R 1.5 time constants after the switch is opened.

(d) When the switch is closed at t = 0, calculate the time constant

(numerical value) for the growth of current in the inductor. Use full

loops and junctions with no short cuts. Use the current designations

given.

1 2 3g = 150 V; R = 125 S; R = 175 S; R = 200 S; L = 18.0 mH

102. For the circuit shown, the switch S is closed at t = 0.

1 3(a) Find the values of I and I at t = 0.

1 3(b) Find the values of I and I at t = 4.

2(c) Find the value of I immediately after the switch is opened.

(d) Using full loops and junctions, calculate both time constants for this

circuit (charging and discharging).

1 2 3g = 150 V; R = 130 S; R = 250 S; R = 350 S; L = 0.35 H

103. A toroid has a rectangular cross section as shown. It has 375 turns.

(a) Calculate the total magnetic flux through the toroid (through the cross hatched

oarea), if the current in each turn is I .

(b) Calculate the self-inductance of the toroid.

104. In the circuit shown the switch is closed at t = 0 after being open for a long time.

(a) Find the current in the inductor at t = 4.

(b) Calculate the current in the inductor after 1.25 time constants have elapsed

from t = 0.

(c) The switch is opened after being closed for 1.25 time constants in (a).

Calculate the time constant for the decay of the current in the inductor.

(d) Calculate the time constant for the growth of the current in the inductor with the

switch closed. You must develop the differential equation governing the

current as function of time. For convenience in grading use the branch current

labels and directions given in the drawing. No shortcuts from other classes.

1 2 3g = 225 V, R = 7,500 S, R = 12,500 S, R = 12,500 S, L = 5.35 mH

105. A toroidal inductor has a rectangular cross section as shown. It has 472 turns.

(a) Calculate the total magnetic flux in the toroid (through the cross-hatched area),

oif the current in each turn is I . Show all details.

(b) Calculate the self-inductance of this toroid.

(c) Calculate the total energy stored between R = a and R = (b+a)/2.

106. The switch in the circuit is closed at t = 0, after being open for a long time.

3(a) Calculate the current in R at t = 0.

1(b) Calculate the current in R at t = 4.

2(c) Calculate the current in R at t = 1.7 J.

(d) Calculate the time constant for discharge when the switch has been

closed a long time and then opened.

(e) Using full loops and junctions as discussed in class, calculate the

charging time constant for the current in L.

1 2 3g = 250 V; R = 75.0 S; R = 127 S; R = 150 S; L = 2.27 mH

107. A 375 pf capacitor is charged to 135 volts. The inductance is 76.0 millihenry.

The switch is closed at t = 0.

(a) Find the peak value of the current in L.

(b) Find the frequency in Hz of the system.

(c) Write an expression for the charge on C as a function of time. Put in all

the correct numerical constants.

(d) Write an expression for the current in L as a function of time. Put in all the

correct numerical constants.

(e) Calculate the total energy stored in the system.

108. For the circuit shown, the switch is closed at t = 0 after being open for a long time.

(a) What is the current in the inductor at t = 4?

3(b) Find the current in R after 1.20 time constants have elapsed.

2(c) Find the current in R after 1.20 time constants have elapsed.

(d) If the switch is closed for a long time and then opened at t = 0, write an

2expression for the current in R as a function of time, with all numerical

quantities evaluated.

(e) Using the current arrows in the diagram, write 3 equations governing the

behavior of this circuit after the switch is closed. Obtain from these the

differential equation governing the current in the inductor, and from this,

calculate the numerical value of the time constant.

1 2 3g = 165 V, R = 750 S, R = 550 S, R = 425 S, L = 3.75 mH

109. Given a long straight wire carrying a current of 6.45 A, calculate the magnetic

energy stored in cylindrical region of space 1.00 cm to 3.00 cm away from the

wire, and 1.32 m long. Numerical answer is needed for full credit.

110. (a) Calculate the magnetic flux through a section of the physics parking that is

6.50 m × 3.25 m. Assume the lot is level and that the earth’s magnetic field is 0.770

gauss and 65° from the horizontal.

(b) Three long, straight wires are perpendicular to the paper. Wire Îhas a current of 1.10 A out of the paper, Ï has a current of 2.30 A

into the paper and Ð has a current of 3.15 A out of the paper.

Calculate the magnitude of the resulting magnetic field at P.

(c) Find the direction of the magnetic field in (b) in the coordinate

system shown.

(d) Calculate the radius of the circular path of protons (m = 1.67 × 10 kg) moving at v = 1.62 × 10 m/s in a-27 5

magnetic field of 3270 gauss.

(e) Calculate the inductance of an ideal solenoid of 3650 turns, length 0.470 m, and circular cross section of

radius 0.012 m. (Numerical answer.)

111. A big, fat, long, straight wire carries a current I, with uniform

ocurrent density. The wire has a radius R .

(a) Calculate the energy stored in the magnetic field between

o oR = R and R = 2R , for a length R of the wire.

(b) Calculate the energy stored in the magnetic field inside the

wire for a length R of the wire.

112. For the circuit shown the switch is open for a long time, and closed at t = 0.

3(a) What is the current in R at t = 0?

3(b) What is the current in R at t = 4?

2(c) Calculate the current in R after 1.20 time constants have elapsed from

t = 0.

(d) Determine the numerical value of the time constant for discharge when the

switch is opened.

(e) Using full loop and junctions and no shortcuts learned in other classes,

calculate the time constant (numerical value) for charging beginning at t = 0.

1R = 150 S g = 135 V

2R = 175 S L = 25.2 mHy

3R = 375 S

113. In a simple circuit involving just an inductor and a capacitor, the charge on the capacitor can be described by Q =

(2.25 × 10 C) cos (8.37 × 10 t + B/3). The value of the capacitance is 475 pF.-7 4

(a) Calculate the inductance, L.

(b) Calculate the maximum potential difference on the capacitor.

(c) Calculate the maximum current in the inductor.

(d) Calculate the total energy stored in the circuit.

(e) Calculate the current in the inductor at t = 0.

114. A 100 pf capacitor is charged to 100 volts. The inductance is 100 millihenry.

The switch is closed at t = 0.

(a) Find the peak value of the current in L.

(b) Find the frequency in Hz of the system.

(c) Write an expression for the charge on C as a function of time. Put in all

the correct numerical constants.

(d) Write an expression for the current in L as a function of time. Put in all the

correct numerical constants.

115. Given the LC circuit shown. At t = 0 the current in the inductance is 0.175 A, in

the direction shown, and the voltage on the capacitor is +17.2 V. C = 3.5 :F,

L = 27.0 mH.

(a) Write an expression for the current in the inductance as a function of time,

getting all the numerical constants and signs right.

(b) Calculate the total energy in the oscillating system.

(c) Calculate the maximum voltage on the capacitor.

116. An LC circuit has an inductance L = 3.0 mh and a capacitance C = 10 :F.

(a) Calculate the angular frequency T of oscillation.

(b) Find the period T of the oscillation.

(c) At time t = 0 the capacitor is charged to 200 :C and the current is zero. Find the value of the maximum

current.

117. The switch S is closed for a long time until all transient behavior has died

out. At t = 0 the switch is opened. g = 1500, R = 1200 S, C = 3.00 :F,

L = 2.85 mH

(a) Find the voltage across the capacitor as a function of time. Evaluate

all numerical constants in this function and express your answer

using these numerical constants including units.

(b) Find the current in the inductor as a function of time. Evaluate all

numerical constants in this expression and express your answer

using these numerical constants including the units.

(c) How much energy is stored in the resonant circuit?

118. In the circuit given, at t = 0 the current I = +1.05 A and the charge on the capacitor Q = +36.2 :C, with the signs

and directions as given in the drawing.

C = 255 pF, L = 374 mH.

(a) Calculate the total energy stored in the system.

(b) Obtain a solution for the charge on the capacitor as a function of time, in the

form Q = D cos(Tt + N). Evaluate completely all numerical constants, with

units, in the resulting expression.

(c) Obtain a solution for current I as a function of time in the same form as in

(b).

119. In the drawing shown, switch S is closed for a long time until all transient

behavior has died out. At t = 0 the switch is opened.

(a) Calculate the voltage across the capacitor as a function of time.

Evaluate all numerical quantities and express your answer using these

quantities, including units.

(b) Calculate the current in the inductor as a function of time, evaluating

all numerical quantities properly.

(c) How much energy is stored in this system?

(d) What is the frequency, in Hz, of the oscillations?

g 100 V, R = 1700 S, C = 7.25 pF, L = 3.27 mH

120. In the circuit shown the switch S is in position 1 for a long time. The switch is

moved to position 2 at t = 0. (The switch is of the type that connects to 2 before

opening at 1.)

(a) Write an expression for the current in the inductance as a function of time.

(b) Write an expression for the charge on the capacitor as a function of time.

121. For the circuit shown the switch is connected at A for a long time. At t = 0 the

switch is opened. The arrow shows the direction for positive current.

(a) Write a complete solution for the current after t = 0 in the form

I = A sin Tt + B cos Tt with all quantities numerically evaluated.

(b) Write a complete solution for the voltage across the capacitor in the

form V = D cos(Tt ! 2) with all quantities numerically evaluated.

(c) What is the maximum energy stored on the

capacitor?

g = 50.0 V; R = 175 S; L = 25.0 :H; C = 4.7 :F

122. For the circuit shown, the direction of positive current is given by the arrow.

At t= 0 the current in the inductance is I = ! 0.245 A, and the charge on the

capacitor is Q = + 1.00 × 10 C. -4

(a) Write a complete solution for the circuit in the form I = D cos(Tt! *)

with D, T and * numerically evaluated.

(b) Write a complete solution for the charge on the capacitor in the form

o oQ = Q cos(Tt ! *) with Q , T and * numerically evaluated. (Yes, I

want the cosine solution.)

(c) Calculate the total energy in the oscillations.

L = 45.0 mH; C = 15.0 × 10 F!6

123. In the circuit shown switch A is closed for a long time. It is then opened and

switch B is closed at t = 0.

(a) Calculate the frequency, f, in Hz AND the period, T, in sec, for the

oscillations.

(b) Write a complete expression for the charge on the capacitor as a

function of time with all numerical values evaluated.

(c) Write a complete expression for the current in the inductance as a

function of time with all numerical values evaluated. Use arrow as the

positive current direction.

= 325 V C = 1.60 × 10 F L = 1.27 × 10 H!6 !3

124. In a simple circuit involving an inductor and a capacitor, the charge on the capacitor can

be described by: Q = (3.75 × 10 C) sin (6.50 × 10 t ! B/6) .!7 5

(t is in seconds.) The value of the capacitance is 325 pF.

(a) Calculate the inductance, L.

(b) Find the maximum potential across the capacitor.

(c) What is the maximum current in the inductor?

(d) Determine the maximum energy stored in the current.

(e) What is the magnitude of the current in the inductor at t = 0?

125. (a) Take the earth's magnetic field as 1.25 × 10 T. Calculate the magnetic energy in a cube 3.00 m on a side-4

at the earth's surface.

(b) Calculate the current in the inductor when 0.0500 s have passed after the switch is closed.

g = 300 V; R = 175 S; L = 14.0 mH

(c) Find the frequency of oscillation in the circuit given if L = 49.0 mH and

C = 475 pF.

(d) If the maximum current in the inductor is 29.0 mA, calculate the maximum voltage on the

capacitor. L = 49.0 mH; C = 475 pF

(e) The earth's magnetic field of 1.20 × 10 T is at an angle of 23° from the horizontal. Calculate-4

the magnetic flux through a circle of radius 2.75 cm on the physics lecture table.

C = 5.55 pF

136. In the circuit shown the positive current is in the direction of the arrow and positive charge is on the capacitor

when the signs are as shown. At t = 0, I = !75.2 mA and Q = +3.25 × 10 C.!9

(a) Determine the frequency of the oscillations.

(b) Calculate the total energy stored in the oscillating system.

(c) Find the maximum voltage on the capacitor.

(d) Write a complete solution for the current in the inductance in the form

I = D cos (Tt - N) with all constants numerically evaluated, including signs.

L = 31.5 :H; C = 29.5 pF

127. In a simple circuit involving only an inductor and a capacitor, the charge on the

capacitor can be written as: [use for parts (a) to (e)]

Q = (4.20 × 10 Coulomb) sin (7.25 × 10 s t)-6 8 -1

(a) Calculate the inductance L.

(b) What is the maximum current in the inductor?

(c) What is the maximum charge on the capacitor?

(d) What is the total energy stored in the circuit?

(e) What is the value of the current in the inductor at t = 0?

128. (a) A simple LC circuit has an inductance of 4.25 mH and a capacitance of 17.2 pF. Calculate

the resonant frequency (in Hertz).

(b) Take the earth’s magnetic field as 0.275 gauss (2.75 × 10 T). Calculate the magnetic energy in a cube-5

5.00 m on a side at the earth’s surface.

(c) If the maximum voltage on the capacitor is 49.0 V, calculate the maximum current in the

inductor. C = 325 pF; L = 75.0 mH.

(d) If the earth’s magnetic field makes the angle shown with the physics lecture table,

calculate the magnetic flux through 6.00 m of that table. Take the magnitude of2

the earth’s field as 0.275 gauss.

(e) A wire carries a current in the direction shown. If the magnitude of the current is decreasing, what is

the direction, clockwise or counter clockwise, of the current in the loop. Both the wire and loop are in

the plane of the paper.

o129. The current in the inductor is given by I = I cos (Tt - B/3). Current in the inductor

upward is positive and when the capacitor is charged as shown, Q is positive.

(a) Calculate an expression for the charge on the capacitor as a function of time.

(b) Find the total energy in the system.

(c) Determine the maximum value of the voltage across the capacitor.

(d) Calculate the first time after t = 0 when the energy is equally divided between the inductor and the

capacitor.

oI = 0.150 A; L = 330 mH; C = 650 pF

Top View Cross Section

130 In a simple circuit involving just an inductor and a capacitor, the charge on the capacitor can be described by Q =

(4.47 × 10 C) cos (2.45 × 10 t + B/5). The value of the capacitance is 475 pF.-6 5

(a) Calculate the inductance, L

(b) Calculate the maximum potential difference on the capacitor.

(c) Calculate the magnitude of the maximum current in the inductor.

(d) Calculate the total energy stored in the circuit.

(e) Calculate the magnitude of the current in the inductor at t = 0.

131. (a) Calculate the reactance of a capacitor of 45.0 pf at a frequency of 175 MHz (1.75 × 10 Hz). 8

(b) Calculate the reactance of an inductor of L = 15.1 mH at an angular frequency of 1.45 × 10 rad/sec. 6

(c) Calculate the charging time constant for the circuit shown if g = 175 V,

R = 750 S, L = 47.5 mH.

(d) For the circuit shown, C = 45.0 pf and L = 76.0 mH. Calculate the frequency of

oscillations, in Hz.

(e) An ideal solenoid has 4755 turns in a length of 1.35 m. It has circular cross section, with a diameter of

3.27 cm. Calculate its self-inductance.

132. A toroid with rectangular cross section as shown has 975 evenly spaced turns of wire and carries a current of 2.75

A.

(a) Calculate the magnetic field inside the toroid at an

arbitrary value of r for a < r < b.

(b) Calculate the magnetic flux through the cross hatched

area shown. It has dimensions w in width and h in

height.

(c) If the border of the cross hatched area is a wire of

resistance 12.0 ohms, calculate a complete expression

for the induced current in it as a function of time if

the current in the toroid is I = (2.75 A) cos Tt.

133. The capacitor shown is charged to 45.0 volts, and then, at t = 0, the switch is closed. C = 700 :F; L = 3.30 mH

(a) Write a complete expression (all possible numerical values evaluated) for the charge on the

capacitor as a function of time.

(b) Write a complete expression (all possible numerical values evaluated) for the current in the

inductor as a function of time.

(c) Calculate the total energy in the oscillating system shortly after t = 0.

(d) After some time has passed, the energy in the oscillating system has diminished to exactly 1/2 its

original value. Calculate the new complete expression (all possible numerical values evaluated) of the

current in the inductor.

134. (a) Take the magnitude of the Earth’s magnetic field as 1.15 × 10 T. If it is at an angle of 18.0° from the!4

horizontal, calculate the magnetic flux through an area on the physics lecture table of 2.2 m ____2

(b) For the value of the Earth’s field given in (a), calculate the magnetic energy in a cube 4.25 m on a side at

the earth’s surface. _______________________________________________________

(c) Calculate the reactance of a capacitor of 65.0 pF at a frequency (f) of 11.2 MHz. ______________

(d) If the maximum current in the inductor is 17.2 mA, calculate the maximum

charge on the capacitor. L = 72.0 mH, C = 320 pF. __________________

(e) A wire carries a current in the direction shown. If the magnitude of the current is

decreasing, what is the direction, clockwise or counter clockwise, of the current in the

loop. Both the wire and loop are in the plane of the paper. ____________________

135. A 275 pF capacitor is charged to 175 volts. the switch is closed at t = 0. L = 950 millihenry.

(a) Find the frequency (f) of the oscillations in the system.

(b) Find the peak value of the current in the inductor.

(c) When the energy is equally divided between L and C, calculate the current in

the inductor.

(d) Write an expression for the charge Q on the capacitor as a function of time.

Put in all numerical constants.

(e) Write an expression for the magnitude of the current in the inductor as a

function of time evaluating all numerical constants.

o136. A big, fat, long, straight wire carries a current I, with uniform current density. The wire has a radius R .

(a) Calculate the energy stored in the magnetic field between

o oR = R and R = 1.5 R , for a length R of the wire.

(b) Calculate the energy stored in the magnetic field inside the

wire for a length R of the wire.

Data: Use these constants (where it states, for example, 1 ft, the 1 is exact for significant figure purposes).

1 ft = 12 in (exact)1 m = 3.28 ft1 mile = 5280 ft (exact)1 hour = 3600 sec = 60 min (exact)1 day = 24 hr (exact)

earthg = 9.80 m/s = 32.2 ft/s2 2

moong = 1.67 m/s = 5.48 ft/s2 2

1 year = 365.25 days1 kg = 0.0685 slug1 N = 0.225 pound1 horsepower = 550 ft@pounds/s (exact)

earthM = 5.98 × 10 kg24

earthR = 6.38 × 10 km3

sunM = 1.99 × 10 kg30

sunR = 6.96 × 10 m8

moonM = 7.35 × 10 kg22

moonR = 1.74 × 10 km3

G = 6.67 × 10 N@m /kg-11 2 2

k = 9.00 × 10 N@m /C9 2 2

og = 8.85 × 10 F/m-12

electron chargee = -1.60 × 10 C-19

electronm = 9.11 × 10 kg-31

o: = 4B × 10 T@m/A (exact)-7

N(Avogadro's No.) = 6.02 × 10 atoms/gm@mole23

= 6.02 × 10 atoms/kg@mole26

1 Tesla = 10,000 gauss (exact)

2D(H O) = 1000 kg/m3

cos(a ± b) = cos a cos b K a sin bsin(a ± b) = sin a cos b ± sin b

protonm = 1.67 × 10 kg!27

B = 3.14