Question

The electric field strength if the potential of field depends upon x, y coordinates as $V=a(x^2-y^2)$ is:

• A. $E=2a\sqrt{x^2+y^2}$
• B. $E=a\sqrt{x^2+y^2}$
• C. $E=\frac{a}{2}\sqrt{x^2+y^2}$
• D. $E=a\sqrt{\frac{1}{x^2+y^2}}$

Answer 1

The correct option is A $E=2a\sqrt{x^2+y^2}$

We know
$E=-\frac{dv}{dx}\left(\hat{i}\right)-\frac{dv}{dy}\left(\hat{j}\right)-\frac{dv}{dz}\left(\hat{k}\right)$
$\Rightarrow =-\frac{d}{dx}\left(a\right(x^2-y^2\left)\right)\hat{i}-\frac{d}{dy}\left(a\right(x^2-y^2\left)\right)\hat{j}$
$\Rightarrow E=-a2x\hat{i}+a2y\hat{j}$
$\Rightarrow E=-2ax\hat{i}+2ay\hat{j}$
$\therefore |E|=\sqrt{(-2ax{)}^2+(2ay{)}^2}$
$\therefore E=2a\sqrt{x^2+y^2}$