Question

In Figure below, two identical resistors of resistance R = 24Ω, and a variable resistor R_x are connected to a battery of voltage V as shown. What should be the value of the variable resistance R_x to make the voltage across the two parallel resistor equal to V/ 8?

Select one:

a. 16 Ω

b. 40 Ω

c. 8 Ω

d. 24 Ω

e. 4 Ω

Answer 1

Answer : (e) 4 Ω

Solution :

Here, Equivalent resistance of the parallel combination will be :

R_P = R R_x / (R + R_x)

Here, Equivalent resistance of the circuit will be :

R_eq = R_P + R = { R R_x / (R + R_x) } + R

So, Current through the circuit will be : I = V / R_eq

And, Potential difference across Resistor R connected in series combination with the battery will be :

V_R = I R = (V / R_eq) R

Since, voltage across the two parallel resistor equal to V/8.

So, V_R = V - V/8 =  7V/8

∴ 7V/8 = (V / R_eq) R

∴ 7/8 = R / R_eq

∴ R_eq = 8R/7  

∴ { R R_x / (R + R_x) } + R = 8R/7

∴ { R R_x / (R + R_x) } = R/7

∴ R R_x = R/7 (R + R_x)

∴ 7R R_x = R (R + R_x)

∴ 7R R_x = R² + RR_x

∴ 6R R_x = R²

∴ 6 R_x = R

∴ R_x = R/6 = (24 Ω) / 6

∴ R_x = 4 Ω