Two identical resistors of resistance R = 24 Ω and a variable resistor R_x are connected to an ideal battery of voltage V as shown in the figure. What should be the value of the variable resistance R_x to make the voltage across the two parallel resistors equal to . A) 4.0 Ω B) 24 Ω C) 8.0 Ω D) 16 Ω E) 40 Ω

Question

Quizplus · Accepted Answer

The two parallel resistors have a combined resistance of 12 Ω ($R_{tot} = \frac{RR}{R+R} = 12$).

Therefore, the total current through the circuit is $I = \frac{V}{R_{tot}} = \frac{V}{12}$.

Since the two parallel resistors have the same resistance, they each have half of the total current, so the current through each of them is $I_1 = I_2 = \frac{V}{24}$.

The voltage drop across each of the parallel resistors is then:

$V_1 = I_1R = \frac{V}{24} \cdot 24 = V/2$

$V_2 = I_2R = V_1$

So we want the voltage drop across the variable resistor to also be $V/2$.

Using the voltage divider formula, we get:

$V_{	ext{variable resistor}} = \frac{R_{	ext{variable resistor}}}{R_{	ext{variable resistor}} + 24} \cdot V$

$V_{	ext{variable resistor}} = \frac{R_{	ext{variable resistor}}}{R_{	ext{variable resistor}} + 24} \cdot \frac{V}{2}$

Solving for $R_{	ext{variable resistor}}$, we get:

$R_{	ext{variable resistor}} = \frac{24}{\frac{V}{2} / V - \frac{1}{2}} = 8 \Omega$

Therefore, the value of the variable resistor should be 8 Ω.

Two Identical Resistors of Resistance R = 24 Ω and a Variable