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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
$6\sqrt5$ N
B
30 N
C
24 N
D
$4\sqrt{35}$ N
Detailed Solution
$\vec E = -\frac{\partial V}{\partial x}\hat i - \frac{\partial V}{\partial y}\hat j - \frac{\partial V}{\partial z}\hat k = -[(6 - 8y)\hat i + (-8x - 8 + 6z)\hat j + (6y)\hat k]$
At (1, 1, 1): $\vec E = 2\hat i + 10\hat j - 6\hat k$
$|\vec E| = \sqrt{2^2 + 10^2 + 6^2} = \sqrt{140} = 2\sqrt{35}$ N/C
$F = qE = 2\times 2\sqrt{35} = 4\sqrt{35}$ N
At (1, 1, 1): $\vec E = 2\hat i + 10\hat j - 6\hat k$
$|\vec E| = \sqrt{2^2 + 10^2 + 6^2} = \sqrt{140} = 2\sqrt{35}$ N/C
$F = qE = 2\times 2\sqrt{35} = 4\sqrt{35}$ N