Question

Electric potential?

please solve it in detail?

Answer 1

(B)

Explanation:

We know that electric potential `V` is related to space charge density `rho` through Poisson's equation

`gradcdot grad V=(-rho)/epsilon_0`
`=>grad^2V=(-rho)/epsilon_0`
where `epsilon_0` is permittivity.

It is given that the potential function is dependent on `x`-coordinate only. Therefore, we get

`rho=-epsilon_0(d^2V)/dx^2`

Inserting given function of `V` we have

`rho=-epsilon_0(d^2)/dx^2(-2x^3+3)`
`=>rho=epsilon_0(d)/dx(6x^2)`
`=>rho=12epsilon_0 x`

.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-

The electric field `vecE` is related to the charge density `rho` through the divergence relationship

`grad cdot vecE=rho/epsilon_0`

and the electric field is related to the electric potential `V` by a gradient relationship

`vecE=-gradV`

Combining these two we get Poisson's equation