In a region, the potential is represented by where is in volts and are in metres. The electric force experienced by a charge of coulomb situated at point (1, 1, 1) is [2014]

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Solution

Q.
In a region, the potential is represented by $V (x, y, z) = 6 x - 8 x y - 8 y + 6 y z,$ 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 [2014]

1 $6 \sqrt{5}$ N

2 $30$ N

3 $24$ N

4 $4 \sqrt{35}$ N

Ans.
(4)

Here, $V (x, y, z) = 6 x - 8 x y - 8 y + 6 y z$

The $x, y$ and $z$ components of electric field are

$E_{x} = - \frac{\partial V}{\partial x} = - \frac{\partial}{\partial x} (6 x - 8 x y - 8 y + 6 y z) = - (6 - 8 y) = - 6 + 8 y$

$E_{y} = - \frac{\partial V}{\partial y} = - \frac{\partial}{\partial y} (6 x - 8 x y - 8 y + 6 y z) = - (- 8 x - 8 + 6 z) = 8 x + 8 - 6 z$

$E_{z} = - \frac{\partial V}{\partial z} = - \frac{\partial}{\partial z} (6 x - 8 x y - 8 y + 6 y z) = - 6 y$

$\vec{E} = E_{x} \hat{i} + E_{y} \hat{j} + E_{z} \hat{k}$

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

At point (1, 1, 1)

$\vec{E} = (- 6 + 8) \hat{i} + (8 + 8 - 6) \hat{j} - 6 \hat{k} = 2 \hat{i} + 10 \hat{j} - 6 \hat{k}$

The magnitude of electric field $\vec{E}$ is

$\vec{E} = \sqrt{E_{x}^{2} + E_{y}^{2} + E_{z}^{2}} = \sqrt{2^{2} + 10^{2} + {(- 6)}^{2}}$

$= \sqrt{140} = 2 \sqrt{35}$ $N$ $C^{- 1}$

Electric force experienced by the charge
$F = q E = 2 C \times 2 \sqrt{35} {N C}^{- 1} = 4 \sqrt{35}$ $N$

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