Question

The electric potential at a point (x, y) in the x-y plane is given by V = - Kxy. The field intensity at a distance r in this plane, from the origin is proportional to :

• A. $r^2$
• B. $r$
• C. $1/r$
• D. $1/r^2$

Answer 1

The correct option is B $r$

Let the $x-y$ coordinates of the point at distance $r$ from the origin be given as $x=rcos\theta$, $y=rsin\theta$

Potential is given as $V=-Kxy$

Now, Electric field intensity is $\overrightarrow{E}=(-dV/dx)\overrightarrow{i}+(-dV/dy)\overrightarrow{j}=K(y\overrightarrow{i}+x\overrightarrow{j})=K(rsin\theta \overrightarrow{i}+rcos\theta \overrightarrow{j})=Kr(sin\theta \overrightarrow{i}+cos\theta \overrightarrow{j})\propto r$

So electric field potential is proportional to $r$.