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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 $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in meters. The electric force experienced by a charge of 2 coulomb situated at point $(1,1,1)$ is-Option A $\quad 6 \sqrt{5} \mathrm{~N}$Option B $\quad 30 \mathrm{~N}$Option C $\quad 24 \mathrm{~N}$Option D $\quad 4 \sqrt{35}$
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 $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in meters. The electric force experienced by a charge of 2 coulomb situated at point $(1,1,1)$ is-
Option A $\quad 6 \sqrt{5} \mathrm{~N}$
Option B $\quad 30 \mathrm{~N}$
Option C $\quad 24 \mathrm{~N}$
Option D $\quad 4 \sqrt{35}$
Physics
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 $\mathrm{x}, \mathrm{y}, \mathrm{z}$ are in meters. The electric force experienced by a charge of 2 coulomb situated at point $(1,1,1)$ is-
Option A $\quad 6 \sqrt{5} \mathrm{~N}$
Option B $\quad 30 \mathrm{~N}$
Option C $\quad 24 \mathrm{~N}$
Option D $\quad 4 \sqrt{35}$

The correct option is D

$\begin{aligned}
&\overrightarrow{\mathrm{E}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}} \hat{\mathrm{i}}-\frac{\partial \mathrm{V}}{\partial \mathrm{y}} \hat{\mathrm{j}}-\frac{\partial \mathrm{V}}{\partial \mathrm{z}} \hat{\mathrm{k}} \\
&\overrightarrow{\mathrm{E}}_{\mathrm{x}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{x}} \hat{\mathrm{i}}=(6-8 \mathrm{y}) \hat{\mathrm{i}} \\
&\overrightarrow{\mathrm{E}}_{\mathrm{y}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{y}} \hat{\mathrm{j}}=(-8 \mathrm{x}-8+6 \mathrm{z}) \hat{\mathrm{j}} \\
&\overrightarrow{\mathrm{E}}_{\mathrm{z}}=-\frac{\partial \mathrm{V}}{\partial \mathrm{z}} \hat{\mathrm{k}}=(6 \mathrm{y}) \hat{\mathrm{k}} \\
&\overrightarrow{\mathrm{E}}=\overrightarrow{\mathrm{E}}_{\mathrm{x}}+\overrightarrow{\mathrm{E}}_{\mathrm{y}}+\overrightarrow{\mathrm{E}}_{\mathrm{z}} \\
&=[(6-8 \mathrm{y}) \hat{\mathrm{i}}(-8 \mathrm{x}-8+6 \mathrm{z}) \hat{\mathrm{j}}+(6 \mathrm{y}) \hat{\mathrm{k}}] \\
&\operatorname{At}(1,1,1), \overrightarrow{\mathrm{E}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}-6 \hat{\mathrm{k}} \\
&=(\overrightarrow{\mathrm{E}})=\sqrt{2^{2}+10^{2}+6^{2}}=\sqrt{140}=2 \sqrt{35} \\
&\text { Force }=\mathrm{qE}=2 \times 2 \sqrt{35}=4 \sqrt{35} \mathrm{~N}
\end{aligned}$

i

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