The electric potential for various points in x-y plane is given by ...

Question

The electric potential for various points in x-y plane is given by V = 1.0 x$_{2}$ – 2.0 y$_{2}$, where V is in volts and x, y are in metres. The angle that the electric field at point (2.0 m, 1.0 m) makes with the positive x-axis is –

Accepted Answer

$V=1.0x^2-2.0y^2$

$$E_x=-\frac{\partial V}{\partial x}=-2x,\qquad E_y=-\frac{\partial V}{\partial y}=4y$$

At $(2.0,1.0)$: $E_x=-2(2)=-4$, $E_y=4(1)=4$

Angle with $+x$-axis: $\tan\theta=\dfrac{E_y}{E_x}=\dfrac{4}{-4}=-1$

Since $E_x<0$, $E_y>0$ (2nd quadrant):

$$\theta=180°-45°=135°$$

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