unit 4: gauss’ law today’s oncepts · gauss’ law always true! in cases with symmetry can pull...
TRANSCRIPT
Unit 4, Slide 1
Physics 2112
Unit 4: Gauss’ Law
Today’s Concepts:
A) Conductors
B) Using Gauss’ Law
Unit 4, Slide 2
Use #1: Determining E without calculus
E from infinite line of charge.
We did this before with calculus.
Remember?
22
2
0)tan*(
sin*sec*)(
hh
hdkPE
f
x
Let’s do it again using Gauss Law
o
liner
E
2
A solid conducting sphere
with net charge –Q and a
radius r1. What is the
electric field within the
sphere some distance r2
from the center? (r2<r1)
-Q
r1r2
A. -kQ/r12
B. -kQ/r22
C. -kQ(r2/r1)2
D. Zero
E. None of the above
Unit 3, Slide 3
Use #2: Determine Charge on Surfaces
Unit 4, Slide 4
E 0 inside any conductor at equilibrium
(If E ≠0, then charge feels force and moves!)
Excess charge on conductor only on surface at equilibrium
Why? Apply Gauss’ Law
E 0
Use #2: Determine Charge on Surfaces
o
enc
surface
QAdE
0
0encQ
Take Gaussian surface to be just inside conductor surface
Unit 4, Slide 5
ALWAYS TRUE!
How Does This Work?
Charges in conductor move to surfaces to make Qenclosed 0.
We say charge is induced on the surfaces of conductors
Gauss’ Law + Conductors + Induced Charges
o
enc
surface
QAdE
Question
Unit 4, Slide 6
An 2m x 3m sheet of plastic
has a surface charge
density s = 6uC/m2.
What is the linear charge density of the plate, λx, in the
x direction?
A) x = 36 μC/m
B) x = 18 μC/m
C) x = 12 μC/m
D) x = 6 μC/m
E) x = 0
2m
3mx
Question
Unit 4, Slide 7
An 3m x 3m sheet of plastic
has a linear charge density in
the x direction of x = 9mC/m.
What is the surface charge density, s, of the sheet?
A) s = 81 mC/m2
B) s = 27 μC/m2
C) s = 9 μC/m2
D) s = 3 μC/m2
E) s = 0
3m
3mx
Unit 4, Slide 8
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
shell of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
Example 4.1
A) What is λb, the linear charge density on the outer
surface of the conducting shell ?
B) What is Ex(R), the electric field at point R, located a
distance dR = 0.9 cm from the origin and making an angle
of 30o with respect to the y-axis as shown?
Question
Unit 4, Slide 9
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
What is λa, the linear charge density on the inner surface of
the conducting shell ?
A) λa = -10.2μC/m
B) λa = -5.1μC/m
C) λa = 0
D) λa = +5.1μC/m
E) λa = +10.2μC/m
Question
Unit 4, Slide 10
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
What is λb, the linear charge density on the outer surface of
the conducting shell ?
A) λb = -10.2μC/m
B) λb = -5.1μC/m
C) λb = 0
D) λb = +5.1μC/m
E) λb = +10.2μC/m
Question
Unit 4, Slide 11
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
What is sa, the surface charge density on the inner surface
of the conducting shell ?
A) sa = +5.1μC/m2
B) sa = (+5.1*2*0.027)μC/m2
C) sa = (+5.1/(2*0.027))μC/m2
D) sa = 0
Question
Unit 4, Slide 12
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
What is sb, the surface charge density on the outter
surface of the conducting shell ?
A) sb = -5.1μC/m2
B) sb = (-5.1/(2*0.050))μC/m2
C) sb = (-5.1/(2*0.027))μC/m2
D) sb = 0
Question
Unit 4, Slide 13
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
A) λ1 only
B) λ1 and λa
C) λa only
D) λ1, λa and λb
Let’s say we wanted to find Ex(R), the electric field at point
R, located a distance dR = 0.9 cm from the origin and
making an angle of 30o with respect to the y-axis as shown.
This value would depend on:
Question
Unit 4, Slide 14
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
cylinder of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
A) λ1 only
B) λ1 and λa
C) λa only
D) λ1, λa and λb
Let’s say we wanted to find Ex(R), the electric field at point
R, located a distance dR = 8 cm from the origin and making
an angle of 30o with respect to the y-axis as shown. This
value would depend on:
Unit 4, Slide 15
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a neutral conducting
shell of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length.
Example 4.1
A) What is λb, the linear charge density on the outer
surface of the conducting shell ?
B) What is Ex(R), the electric field at point R, located a
distance dR = 0.9 cm from the origin and making an angle
of 30o with respect to the y-axis as shown?
Unit 4, Slide 16
An infinite line of charge with linear
density λ1 = -5.1μC/m is positioned
along the axis of a thick conducting
shell of inner radius a = 2.7 cm and
outer radius b = 5.0 cm and infinite
length. The conducting shell is
uniformly charged with a linear charge
density λ 2 = +2.0 μC/m.
Example 4.2(a)
C) What is Ex(R), the electric field at point R, located a
distance dR = 7.0 cm from the origin and making an angle
of 30o with respect to the y-axis as shown?
x R
Unit 4, Slide 17
A hollow insulating spherical shell has
a total charge of Q=-8uC and a
uniform charge distribution. The shell
has an inner radius of a = 2.7 cm and
outer radius b = 5.0 cm
Example 4.3
C) What is |E|, the electric field at point R, located a
distance dR = 4.0 cm from the center of the sphere?
x R
Unit 4, Slide 18
CheckPoint: Charged Conducting Sphere & Shell 1
A positively charged solid conducting sphere is contained within a negatively charged conducting spherical shell as shown. The magnitude of the total charge on each sphere is the same. Which of the following statements best describes the electric field in the region between the spheres?
A. The field points radially outwardB. The field points radially inwardC. The field is zero
Unit 4, Slide 19
CheckPoint: Slightly changed
A positively charged solid conducting sphere is contained within a positively charged conducting spherical shell as shown. The magnitude of the total charge on each sphere is the same. Which of the following statements best describes the electric field in the region between the spheres?
A. The field points radially outwardB. The field points radially inwardC. The field is zero
+q
Unit 4, Slide 21
Gauss’ Law Symmetries
ALWAYS TRUE!
In cases with symmetry can pull E outside and get
Spherical Cylindrical Planar
0A
QE enc
24 rA
0
24 r
QE enc
rLA 2
02
rE
22 rA
02
sE
0
encQAdE
Quick Review: Infinite Sheet of Charge
Unit 4, Slide 22
Unit 4, Slide 23
Gauss’ Law
ALWAYS TRUE!
In cases with symmetry can pull E outside and get
In General, integral to calculate flux is difficult…. and not useful!
To use Gauss’ Law to calculate E, need to choose surface carefully!
1) Want E to be constant and equal to value at location of interest
OR
2) Want E dot A 0 so doesn’t add to integral
0A
QE enc
0
encQAdE
Electricity & Magnetism Lecture 4, Slide 24
CheckPoint: Gaussian Surface Choice
You are told to use Gauss' Law to calculate the electric field at a distance R away from a charged cube of dimension a. Which of the following Gaussian surfaces is best suited for this purpose?
A. a sphere of radius R+1/2a
B. a cube of dimension R+1/2a
C. a cylinder with cross sectional radius of R+1/2a and arbitrary length
D. This field cannot be calculated using Gauss' law
E. None of the above
Electricity & Magnetism Lecture 4, Slide 25
CheckPoint: Charged Sphericlal Shell
A charged spherical insulating shell has inner radius a and outer radius b. The charge density on the shell is ρ. What is the magnitude of the E-field at a distance r away from the center of the shell where r < a?
A. ρ/εo
B. zeroC. ρ(b3-a3)/(3εor2)D. none of the above
Electricity & Magnetism Lecture 4, Slide 26
Example 4.3
A charged spherical insulating shell has inner radius a and outer radius b. The charge density on the shell is ρ.
What is the magnitude of the E-field at a distance r away from the center of the shell where a< r < b?
Electricity & Magnetism Lecture 4, Slide 27
Example 4.4
Point charge 3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.
What is E for the three regions of: r < r1
r1< r < r2
r2 < r ?
neutral conductor
r1
r2
y
x3Q
Electricity & Magnetism Lecture 4, Slide 28
Calculation
Point charge 3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.A) What is E everywhere?
We know: magnitude of E is fcn of rdirection of E is along
neutral conductor
r1
r2
y
x3Q r̂
r < r1
24 rEAdE
QQenc 3
r > r2
2
0
3
4
1
r
QE
2
20 )(
3
4
1
rr
QE
0E
A)
B)
C)
r1 < r < r2
2
0
3
4
1
r
QE
2
10
3
4
1
r
QE
0E
A)
B)
C)
We can use Gauss’ Law to determine EUse Gaussian surface = sphere centered on origin
0
encQAdE
2
0
3
4
1
r
QE
Electricity & Magnetism Lecture 4, Slide 29
Calculation
Point charge 3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.
What is E everywhere?
r < r1
r1 < r < r2
r > r2
2
0
3
4
1
r
QE
0E
neutral conductor
r1
r2
y
x3Q
0
encQAdE
Question
Electricity & Magnetism Lecture 4, Slide 30
Point charge 3Q at center of neutral conducting shell of inner radius r1 and outer radius r2.
A)
B)
C)
D)
What can we say about the relative magnitudes of s1 and s2?
|||| 21 ss <
neutral conductor
r1
r2
y
x3Q
2
1
14
3
r
Q
s
0
encQAdE
|||| 21 ss
|||| 21 ss >
0|| 2 s
Question
Unit 4, Slide 31
Now suppose we give conductor a charge of Q.
What is the surface charges at r1?
r1
r2
+2Q
3Q
chargedconductor
r1
r2
y
x3Q
-Q
2
1
14
3
r
Q
s
2
1
14 r
Q
s
2
1
14
2
r
Q
s
A)
B)
C)
Question
Unit 4, Slide 32
Now suppose we give conductor a charge of Q.
What is the surface charges at r2?
r1
r2
+2Q
3Q
chargedconductor
r1
r2
y
x3Q
-Q
2
2
24
3
r
Q
s
2
2
24 r
Q
s
2
2
24
2
r
Q
s
A)
B)
C)
A positive charge is kept (fixed) at the center inside a fixed
spherical neutral conducting shell. Which of the following
represents the charge distribution on the inner and outer walls
of the shell?
A. B. C. D.
Question
The positive charge is now moved and kept fixed off-center
inside the fixed spherical neutral conducting shell. Which of the
following represents the charge distribution on the inner and
outer surfaces of the shell?
A. B. C.
D. E.
Question
The positive charge +Q is now kept fixed at the center of a
spherical neutral conducting shell. A negative charge –Q is
brought near the outside of the sphere. Which of the following
represents the charge distributions?
A. B.
C.D.
Question