Electric potential?

please solve it in detail?enter image source here

1 Answer
Sep 11, 2017

(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