Example 18.3 Equivalent Resistance Goal Solve a problem involving both series an

Question

Example 18.3 Equivalent Resistance Goal Solve a problem involving both series and parallel resistors.

Problem Four resistors are connected as shown in Figure 18.11a.

(a) Find the equivalent resistance between points a and c.

(b) What is the current in each resistor if a 42 V battery is connected between a and c?

Strategy Reduce the circuit steps, as shown in Figures 18.11 b and 18.11 c, using the sum rule for resistors in series and the reciprocal-sum rule for resistors in parallel. Finding the currents is a matter of applying Ohm's law while working backwards through the diagrams.
Figure 18.11 (Example 18.3) The four resistors shown in (a) can be reduced in steps to an equivalent 14 resistor. Solution (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 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!)

Req = 2.0 In the new diagram, 18.11b, there are now two resistors in series. Combine them with the sum rule to find the equivalent resistance of the circuit. Req = R1 + R2 = 12 + 2.0 =   (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 8.11c, which is the total current. Resistors in series all carry the same current, so this is the current in the 12 in Figure 8.11b, and also in the 8.0 and 4.0 resistors in Figure 8.11a. =  A Apply the junction rule to point b. (1) I = I1 + I2 The 6.0 and 3.0 resistors are in parallel, so the voltage drops across them are the same. V6 = V3 (6.0 )I1 = (3.0 )I22.0 I1 = I2 Substitute this result into Equation (1), with I = 3.0 A. 3.0 A = I1 + 2I1 = 3I1I1 = A

I2 =  A
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, as expected.

Explanation / Answer

resistor 2 ohm and 4 ohm are in parallel.

equivalent resistance of both = 2*4/(2+4) = 4/3 ohm

now equivalent reistance between a and c = 3 + 6 + 4/3 = 10.33 ohm

35 volt battery is connected between a and c.

current through battery = 35/10.33 = 3.39 amp

so value of I = 3.39 amp

now current in 2 ohm = 4*3.39/(2+4) = 2.26 amp = value of I1

current in 4 ohm = 2*3.39/(2+4) = 1.13 amp = value of I2

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