First slide

Electrostatic potential

Question

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

Difficult

A

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

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

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

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

Solution

The electric field $\vec{E}$ and potential V in a region are related as
$\vec{E} = - [\frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k}]$
$Here, V (x, y, z) = 6 xy - y + 2 yz$
$∴ \vec{E} = - [\frac{\partial}{\partial x} (6 xy - y + 2 yz) \hat{i} + \frac{\partial}{\partial y} (6 xy - y + 2 yz) \hat{j}$ $+ \frac{\partial}{\partial z} (6 xy - y + 2 yz) \hat{k}]$
$= - [(6 y) \hat{i} + (6 x - 1 + 2 z) \hat{k} + (2 y) \hat{k}]$
$At point (1, 1, 0)$
$\vec{E} = - [(6 (1)) \hat{i} + (6 (1) - 1 + 2 (0)) \hat{j} + (2 (1)) \hat{k}]$
$= - (6 \hat{i} + 5 \hat{j} + 2 \hat{k})$

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