GATE 2026 IN – Question 15

Electricity and Magnetism · Electrostatics and capacitance · 1 mark · MTA

An electric field has a potential $V(x,y,z)=\sqrt{x^2+y^2+z^2}\ \mathrm{V}$. A charge of 1 coulomb placed at $(\hat{i}+\hat{j}+\hat{k})$ experiences a force $\vec{F}=(a\hat{i}+b\hat{j}+c\hat{k})\ \mathrm{N}$. The values of $(a,b,c)$ are ________.

$a=\frac{1}{\sqrt3},\ b=\frac{1}{\sqrt3},\ c=\frac{1}{\sqrt3}$
$a=\frac13,\ b=\frac13,\ c=\frac13$
$a=0,\ b=1,\ c=\frac{1}{\sqrt3}$
$a=0,\ b=1,\ c=\frac13$

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Correct answer: MTA

Explanation

The electric field is $\vec E=-\nabla V=-(x\hat i+y\hat j+z\hat k)/\sqrt{x^2+y^2+z^2}$.

At $(1,1,1)$, multiplying by the positive charge $q=1\ \mathrm C$ gives $\vec F=-(\hat i+\hat j+\hat k)/\sqrt3\ \mathrm N$. Thus $a=b=c=-1/\sqrt3$.

None of the printed choices gives these negative components. The official final key awards **marks to all**, which is retained.