A potentiometer wire of 400\ cm length has a resistance of 20\ . It...

Total resistance of the circuit is $R_{total} = R_w + R_s = 20 + 15 = 35\ \Omega$.

Current in the circuit is $I = \dfrac{E}{R_{total}} = \dfrac{6}{35}\ \text{A}$.

Potential difference across the potentiometer wire is $V_w = I R_w = \dfrac{6}{35} \times 20 = \dfrac{24}{7}\ \text{V}$.

Length of the potentiometer wire is $L = 400\ \text{cm} = 4\ \text{m}$.

Potential gradient is $k = \dfrac{V_w}{L} = \dfrac{\dfrac{24}{7}}{4} = \dfrac{6}{7}\ \text{V/m}$.

Given $k = \dfrac{x}{7}\ \text{V/m}$, comparing the two expressions yields $x = 6$.

Answer: $6$