(a) Find the equivalent resistance of the circuit. The 8.0- and 4.0- resistors a

Question

(a) Find the equivalent resistance of the circuit.

The 8.0- and 4.0- resistors are in series, so use the sum rule to find the equivalent resistance between a and b.

Req = R1 + R2 = 8.0 + 4.0 = 12.0

The 6.0- and 3.0- resistors are in parallel, so use the reciprocal-sum rule to find the equivalent resistance between b and c (don't forget to invert!).

In the new diagram, figure b, there are now two resistors in series. Combine them with the sum rule to find the equivalent resistance of the circuit.

(b) Find the current in each resistor if a 42-V battery is connected between points a and c.

Find the current in the equivalent resistor in figure c, which is the total current. Resistors in series all carry the same current, so the value is the current in the 12- resistor in figure b and also in the 8.0- and 4.0-resistors in figure a.

I =

=

= 3.0 A

Calculate the voltage drop Vparaacross the parallel circuit, which has an equivalent resistance of 2.0 .

Vpara = IR = (3.0 A)(2.0 ) = 6.0 V

Apply Ohm's law again to find the currents in each resistor of the parallel circuit.

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REMARKS As a final check, note that

Vbc = (6.0 )I1 = (3.0 )I2 = 6.0 V

and

Vab = (12 )I1 = 36 V;

therefore,

Vac = Vab + Vbc = 42 V,

as expected.

QUESTION Which of the original resistors dissipates energy at the greatest rate?

the 8.0 resistorthe 4.0 resistor     the 6.0 resistorthe 3.0 resistor

EXERCISEHINTS:  GETTING STARTED  |  I'M STUCK!

Suppose the series resistors in the example are now 2.00 and 1.00 while the parallel resistors are 2.00 (top) and 1.00 (bottom), and the battery provides an emf of 80.0 V.(a) Find the equivalent resistance.

Req

=  

(b) Find the currents

I, I1, and I2.

I

I1

I2

1 Req = 1 R1 + 1 R2 = 1 6.0 + 1 3.0 = 1 2.0 Req = 2.0

Explanation / Answer

The formula for energy dissipated = I2R
I is the current flowing through a particular resistor R
For 8 resistor, E = 32 x 8 = 72 J/s
For 4 resistor, E = 32 x 4 = 36 J/s.
For 6 resistor, E = 12 x 6 = 6 J/s.
For 3 resistor, E = 22 x 3 = 12 J/s.

So the greatest rate is for 8 .

a)

2 and 1 are in series, their equivalent resistance = 3 .
2 and 1 are in parallel, their equivalent resistance = 2/3 .
Total resistance = 3 + 2/3 = 11/3 = 3.667 .

b)

I = V/Equivalent resistance = 80 / 3.667 = 21.82 A

I1 = 21.82 x (R1/(R1 + R2)) = 21.82 x (1/3) = 7.27 A.

I2 = 21.82 x (R2/(R1 + R2)) =21.82 x (2/3) = 14.545 A

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