In a region, the potential is represented by V (x, y, z) = 6x − 8xy − 8y + 6yz, where V is in volts and x, y, z are in metres. The electric force experienced by a charge of 2 coulomb situated at point (1, 1, 1) is ?

In a region, the potential is represented by V (x, y, z) = 6x − 8xy − 8y + 6yz, where V is in volts and x, y, z are in metres. The electric force experienced by a charge of 2 coulomb situated at point (1, 1, 1) is ?

(a) \( 6\sqrt{5} \ \text{N} \)

(b) \( 30 \ \text{N} \)

(c) \( 24 \ \text{N} \)

(d) \( 4\sqrt{35} \ \text{N} \)

Answer :  (d) \( 4\sqrt{35} \ \text{N} \)

\( \vec{F} = q \vec{E} \)

\( = 2 \times \sqrt{140} \)

\( = 2 \times \sqrt{4 \times 35} \)

\( = 4\sqrt{35} \)

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

\( = -\left[ (6 - 8y) \hat{i} + (8x - 8) \hat{j} + 64y \hat{k} \right] \)

\( = (8y - 6) \hat{i} + (8x + 8) \hat{j} - 64y \hat{k} \)

\( = 2 \hat{i} + 10 \hat{j} - 6 \hat{k} \)

\( |\vec{E}| = \sqrt{2^2 + 10^2 + (-6)^2} \)

\( = \sqrt{4 + 100 + 36} = \sqrt{140} \)