Electrostatic Potential And Capacitance Ques 2
2. 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 metres. The electric force experienced by a charge of $2$ coulomb situated at point $(1,1,1)$ is :
[2014]
(a) $6 \sqrt{5} \mathrm{N}$
(b) $30 \mathrm{N}$
(c) $24 \mathrm{N}$
(d) $4 \sqrt{35} \mathrm{N}$
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Answer:
Correct Answer: 2.(d)
Solution: (d) $\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}} $
$ \text { where } V(\mathrm{x}, \mathrm{y}, \mathrm{z})=6 \mathrm{x}-8 \mathrm{xy}-8 \mathrm{y}+6 \mathrm{yz} $
$ =-[(6-8 \mathrm{y}) \hat{\mathrm{i}}+(-8 \mathrm{x}-8+6 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}} $
$ \Rightarrow \quad(\overrightarrow{\mathrm{E}})=\sqrt{(2)^2+(10)^2+(-6)^2}=\sqrt{140}=2 \sqrt{35} $
$ \therefore \quad \mathrm{F}=\mathrm{q} \overrightarrow{\mathrm{E}}=2 \times 2 \sqrt{35}=4 \sqrt{35}$
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