Question

The electric potential at a point $(x, y)$ in the $x y$ -plane is given by $V=-K x y_{i}$ The electric field intensity at a distance $r$ from the origin varies as

• A. $r^{2}$
• B. r
• C. 2 r
• D. $2 r^{2}$

Answer 1

Answer: (B)

Co-ordinates of the point $=(x, y)$.
Electric potential $(V)=-K x y$
Distance of the point from origin $(r)=\sqrt{x^{2}+y^{2}}$
$E_{x} =-\frac{\partial V}{\partial x}=-\frac{\partial}{\partial x}(-K x y)=K y $
$E_{y} =-\frac{\partial V}{\partial y}=-\frac{\partial}{\partial y}(-K x y)=K x $
$\therefore E_{r} =\sqrt{E_{x}^{2}+E_{y}^{2}}$
$=\sqrt{(K y)^{2}+(K x)^{2}}=K r $
or $ E_{r} \propto r$