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

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

1. A

$-\left(2\stackrel{^}{\mathrm{i}}+3\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)$

2. B

$-\left(6\stackrel{^}{\mathrm{i}}+9\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)$

3. C

$-\left(3\stackrel{^}{\mathrm{i}}+5\stackrel{^}{\mathrm{j}}+3\stackrel{^}{\mathrm{k}}\right)$

4. D

$-\left(\stackrel{^}{6\mathrm{i}}+5\stackrel{^}{\mathrm{j}}+2\stackrel{^}{\mathrm{k}}\right)$

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### Solution:

The electric field $\stackrel{\to }{E}$ and potential V in a region are related as

$\stackrel{\to }{\mathrm{E}}=-\left[\frac{\partial \mathrm{V}}{\partial \mathrm{x}}\stackrel{^}{\mathrm{i}}+\frac{\partial \mathrm{V}}{\partial \mathrm{y}}\stackrel{^}{\mathrm{j}}+\frac{\partial \mathrm{V}}{\partial \mathrm{z}}\stackrel{^}{\mathrm{k}}\right]$

$+\frac{\partial }{\partial \mathrm{z}}\left(6\mathrm{xy}-\mathrm{y}+2\mathrm{yz}\right)\stackrel{^}{\mathrm{k}}]$

$=-\left[\left(6\mathrm{y}\right)\stackrel{^}{\mathrm{i}}+\left(6\mathrm{x}-1+2\mathrm{z}\right)\stackrel{^}{\mathrm{k}}+\left(2\mathrm{y}\right)\stackrel{^}{\mathrm{k}}\right]$

$\stackrel{\to }{\mathrm{E}}=-\left[\left(6\left(1\right)\right)\stackrel{^}{\mathrm{i}}+\left(6\left(1\right)-1+2\left(0\right)\right)\stackrel{^}{\mathrm{j}}+\left(2\left(1\right)\right)\stackrel{^}{\mathrm{k}}\right]$

$=-\left(6\stackrel{^}{\mathrm{i}}+5\stackrel{^}{\mathrm{j}}+2\stackrel{^}{\mathrm{k}}\right)$

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