In the given network, if VA - VC = 8 V, the value of R is

The correct answer is Option (3) → 4.5 Ω

The network is a single loop, so the same current I flows through every element. In the figure the longer plate of each cell is its positive terminal: the 10 V cell has its positive terminal towards B and the 5 V cell towards A. The two cells oppose each other round the loop, so the net emf is 10 V − 5 V = 5 V, in the sense of the 10 V cell. The current leaves B, flows through the 3 Ω resistor to A, through the 5 V cell to D and through R to C.

    \( \displaystyle I = \frac{10 - 5}{3 + R} \)

Now find \( V_A - V_C \) along the path A–B–C. Current flows from B to A in the 3 Ω resistor, so B is 3I higher than A; crossing the 10 V cell from its positive terminal (B) to its negative terminal (C) the potential falls by 10 V.

    \( V_B = V_A + 3I \)
    \( V_C = V_B - 10 = V_A + 3I - 10 \)

    \( V_A - V_C = 10 - 3I = 8 \)
    \( I = 2/3\,\mathrm{A} \)

    \( \displaystyle 3 + R = \frac{5}{I} = \frac{5}{2/3} = 7.5 \)

    \( \displaystyle R = 4.5\,\Omega \)

Check along A–D–C: \( V_A - V_C = 5 + IR = 5 + \frac{2}{3} \times 4.5 = 8 \) V.