PHY-2049

Chapter 27 Circuits

A closed circuit

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Power in DC Circuit

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#24 chapter 26: The figure below gives the electrical potential V(x) along a copper wire carrying a uniform current, from a point at higher potential (x=0m) to a

point at a lower potential (x=3m). The wire has a radius of 2.45 mm. What is the current in the wire?

copper

12 uvolts 0 volts

What does the graph tell us??

*The length of the wire is 3 meters.*The potential difference across the

wire is 12 volts.*The wire is uniform.

Let’s get rid of the mm radius and convert it to area in square meters:A=r2 = 3.14159 x 2.452 x 10-6 m2

orA=1.9 x 10-5 m 2

Material is Copper so resistivity is (from table) = 1.69 x 10-8 ohm meters

We have all what we need….

mA 49.41067.2

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0.3m-ohm 1069.1

3

6

5

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ohms

volts

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mx

A

LR

Let’s add resistors …….

Series CombinationsR1 R2

i i

V1 V2V

iiRseriesR

general

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and

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SERIES Resistors

The rod in the figure is made of two materials. The figure is not drawn to scale. Each conductor has a square cross section 3.00 mm on a side. The first material has a resistivity of 4.00 × 10–3 Ω · m and is 25.0 cm long, while the second material has a resistivity of 6.00 × 10–3 Ω · m and is 40.0 cm long. What is the resistance between the ends of the rod?

Parallel Combination??

R1, I1

R2, I2

V

i iRR

general

RRR

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R

V

R

V

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What’s This???

#26 chapter 27:In Figure below, find the equivalent resistance between points (a) F and H and [2.5]  (b) F and G. [3.13]  

Power Source in a Circuit

The ideal battery does work on charges moving them (inside) from a lower potential to one that is V higher.

A REAL Power Sourceis NOT an ideal battery

V

ε or Emf is an idealized device that does an amount of work to move a unit charge from one side to another.

By the way …. this is called a circuit!

Internal Resistance

A Physical (Real) Battery

Internal Resistance Rr

Emfi

Back to which is brighter?

Back to Potential

Represents a charge in space

Change in potential as one circuitsthis complete circuit is ZERO!

Consider a “circuit”.

This trip around the circuit is the same as a path through space.

THE CHANGE IN POTENTIAL FROM “a” AROUND THE CIRCUIT AND BACK TO “a” is ZERO!!

To remember

In a real circuit, we can neglect the resistance of the wires compared to the resistors. We can therefore consider a wire in a circuit to

be an equipotential – the change in potential over its length is slight compared to that in a resistor

A resistor allows current to flow from a high potential to a lower potential.

The energy needed to do this is supplied by the battery.

VqW

LOOP EQUATION The sum of the voltage drops (or rises)

as one completely travels through a circuit loop is zero.

Sometimes known as Kirchoff’s loop equation.

NODE EQUATION The sum of the currents entering (or

leaving) a node in a circuit is ZERO

Take a trip around this circuit.

Consider voltage DROPS:

ε-ir -iR = 0or

ε=ir + iR

Circuit Reduction

i=ε/Req

Multiple Batteries

Reduction

Computes i

Another Reduction Example

PARALLEL

1212

1

600

50

30

1

20

11

RR

RC Circuit

Initially, no current through the circuit

Close switch at (a) and current begins to flow until the capacitor is fully charged.

If capacitor is charged and switch is switched to (b) discharge will follow.

Really Close the Switch

RRC

q

dt

dq

orC

q

dt

dqR

C

qiR

dt

dqi since

0

Equation Loop

This is a differential equation.

To solve we need what is called a particular solution as well as a general solution.

We often do this by creative “guessing” and then matching the guess to reality.

Result q=Cε(1-e-t/RC)

q=Cε(1-e-t/RC) and i=(Cε/RC) e-t/RC

RCteR

i /

Discharging a Capacitor

qinitial=Cε BIG SURPRISE! (Q=CV)i

iR+q/C=0

RCt

RCt

eRC

q

dt

dqi

eqq

solutionC

q

dt

dqR

/0

/0

0