Question

Consider the charge profile shown in the figure. The resultant potential distribution is best described by

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Ans : (d)

Applying Poisson's equations

${\nabla }^{2}V=\frac{{\partial }^{2}V}{\partial x^2}=-\frac{{\rho }_v}{\in }=K$
Constant charge density

$\frac{\partial V}{\partial x}=-Kx+K^{\prime }$

$V=\frac{-K{x}^2}{2}+K^{\prime }x+K^{\prime \prime }$

Towards positive x or negative side. It is second order parabolic increase. Due to symmetry of + and - charges $K^{\prime \prime }$ = 0 is expected with V = 0 at centre and graph passing through origin.
Beyond x > a or x < b, E = 0 due to capacitive nature of + and - charges.

$V=-\int 0\cdot \overrightarrow{dt}=Cons\tan t$
This constant is same at x = a or x =b.