In a region, the potential is represented by $\mathrm{V}(\mathrm{x}, \mathrm{y}, \mathrm{z})=6 \mathrm{x}-8…

In a region, the potential is represented by $\mathrm{V}(\mathrm{x}, \mathrm{y}, \mathrm{z})=6 \mathrm{x}-8 \mathrm{xy}-8 \mathrm{y}+6 \mathrm{yz}$, where $\mathrm{V}$ is in volts and $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in metres. The electric force experienced by a charge of 2 coulomb situated at $\operatorname{point}(1,1,1)$ is :
  1. $6 \sqrt{5} \mathrm{~N}$
  2. $30 \mathrm{~N}$
  3. $24 \mathrm{~N}$
  4. $4 \sqrt{35} \mathrm{~N}$

Solution

$\mathrm{V}_{(x, y, x)}=6 \mathrm{x}-8 \mathrm{xy}-8 \mathrm{y}+6 \mathrm{yz}$
$\mathrm{E}_{\mathrm{x}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}}=-6+8 \mathrm{y}$
$\mathrm{E}_{\mathrm{y}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{y}}=8 \mathrm{x}+8-6 \mathrm{z}$
$\mathrm{E}_{z}=-\frac{\partial \mathrm{V}}{\partial \mathrm{z}}=-6 \mathrm{y}$
$\overrightarrow{\mathrm{E}}=(-6+8 \mathrm{y}) \hat{\mathrm{i}}+(8 \mathrm{x}+8-6 \mathrm{z}) \hat{\mathrm{j}}-6 \mathrm{y} \hat{\mathrm{k}}$
$\mathrm{E}_{(1,1,1)}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$
$\overrightarrow{\mathrm{|E|}} =2 \sqrt{35} \mathrm{NC}^{-1}$
$\mathrm{~F}=\mathrm{qE}=2 \times 2 \sqrt{35}=4 \sqrt{35 \mathrm{~N}}$ ~

Asked in: JEE Mains - Electrostatics - Test 4

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