The space between the two large parallel plates is filled with a material of uniform charge density `rho`. Assume that one of the plate is kept at `x=0`. The potential at any point x between these plates is given by (A and B are constants).

Options

A. `- (rho x^(3))/(2 in_(0))`
B. `-((rho x^(2))/(2 in_(0))+Ax)`
C. `-((rho x^(2))/(2 in_(0))+Ax+B)`
D. `-((rho x^(3))/(4 in_(0))+Ax^(2)+Bx)`

Answer

C

Explanation

According to the question, the space between the two large parallel plates is filled with a material of uniform charge density `rho` i.e., It is a homogeneous medium, therefore the potential at any point between these plates is calculated by Poission's equation as given below,
`V^(2)V=- rho/epsi_(0)` ...(i)
Since, we have to calculate the potential at any point along x-direction, hence the derivative of the potential along y and z directions is zero.
Hence, from Eq. (i), we get
`(d^(2)V)/(d x^(2))=-rho/epsi_(0)`
Integrate on both the sides, we get
`(d^(2)V)/(d x^(2))=-rho/epsi_(0)`
Integrate on both the sides, we get
`implies int (d^(2) V)/( d x^(2)) dx =-int rho/epsi_(0) dx implies (d V)/(d x)=(-rho x)/epsi_(0) -A`
Integrate on both sides again, we get
`implies int (d V)/(d x) dx =int (- rho x)/epsi_(0) dx - int A dx`
`V=(-rhox^(2))/(2 epsi_(0))-Ax-B`
`:. V=- ((+ rho x^(2))/(2 epsi_(0))+Ax+B]`