Question

The potential in an electric field varies as $V=\left(x^2-y^2\right)$.
The electric lines of the force in $X-Y$ plane are

• A.
• B.
• C.
• D.

Answer 1

Answer: (C)

The potential in an electric field varies as,
$V=\left(x^2-y^2\right)$
$\therefore$ Electric field,
$E=-\left[\frac{dv}{dx}\hat{i}+\frac{dv}{dy}\hat{j}\right][\because E=\Delta V]$
$=-\left[\frac{d}{dx}\left(x^2-y^2\right)\hat{i}+\frac{d}{dy}\left(x^2-y^2\right)\hat{j}\right]$
$E=-[2x\hat{i}-2y\hat{j}]\Rightarrow E=-2x\hat{i}+2\hat{j}$
Expression of the electric field is linear equation in two variables, i.e. straight lines in $X$ - $Y$ plane and slope having $45^{\circ },35^{\circ }$ etc.