Question

The electrostatic potential $V$ at any point (x, y, z) in space is given by $V=4x^2$

• A. The y-and z-components of the electrostatic field at any point are zero.
• B. The x-component of electric field an any point is given by $(-8x\hat{i})$
• C. The x-component of electric field at $(1,0,2)$ is $(-8\hat{i})$
• D. The y-and z-components of the field are constant in magnitude.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A The y-and z-components of the electrostatic field at any point are zero.
B The x-component of electric field an any point is given by $(-8x\hat{i})$
C The x-component of electric field at $(1,0,2)$ is $(-8\hat{i})$
D The y-and z-components of the field are constant in magnitude.

We have $V=4x^2$

So, the $x,y$ and $z$ components of the electrostatic field are

$E_x=\frac{-\partial V}{\partial x}=-8x$

$E_y=\frac{-\partial V}{\partial y}=0$

$E_z=\frac{-\partial V}{\partial z}=0$

So, $\overrightarrow{E}=E_x\hat{i}+E_y\hat{j}+E_z\hat{k}=-8x\hat{i}$.

The electrostatic field at $(1,0,2)$ is $\overrightarrow{E}=(-8)\hat{i}V/m$.