If potential (in volts) in a region is expressed as V (x, y, z) = 6xy − y + 2yz, the electric field (in N/C ) at point (1, 1, 0) is ?

If potential (in volts) in a region is expressed as V (x, y, z) = 6xy − y + 2yz, the electric field (in N/C ) at point (1, 1, 0) is ?

(a) \( - (6\hat{i} + 5\hat{j} + 2\hat{k}) \)

(b) \( - (2\hat{i} + 3\hat{j} + \hat{k}) \)

(c) \( - (6\hat{i} + 9\hat{j} + \hat{k}) \)

(d) \( - (3\hat{i} + 5\hat{j} + 3\hat{k}) \)

Answer :  (a) \( - (6\hat{i} + 5\hat{j} + 2\hat{k}) \)

\( \vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \)

\( = -\left[ 6y \hat{i} + (6x - 1 + 2z) \hat{j} + 2y \hat{k} \right] \)

\( = -\left( 6 \hat{i} + 5 \hat{j} + 2 \hat{k} \right) \)