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,
$E =-\left[\frac{d v}{d x} \hat{ i }+\frac{d v}{d y} \hat{ j }\right] [\because E=\Delta V]$
$=-\left[\frac{d}{d x}\left(x^{2}-y^{2}\right) \hat{ i }+\frac{d}{d y}\left(x^{2}-y^{2}\right) \hat{ j }\right] $
$E =-[2 x \hat{ i }-2 y \hat{ j }] \Rightarrow E=-2 x \hat{ i }+2 \hat{ j }$
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.