A uniformly distributed space charge fills up the space between two large parallel plates separated by a distance `d`. The potential difference between the plates is equal to zero ? What will then be the field strength near the other plate ?

Explanation

In the space between the plates we have the Poission equation
`(del^(2) varphi)/(del x^(2)) = -(rho_(0))/(epsilon_(0))`
or, `varphi = -(rho_(0))/(2 epsilon_(0)) x^(2) + Ax + B`
where `rho_(0)` is the constant space charge density between the plates.
We can choose `varphi (0) = 0` so `B = 0`
Then `varphi(d) = Delta varphi = Ad - (rho_(0) d^(2))/(2 epsilon_(0))` or, `A = (Delta varphi)/(d) + (rho_(0) d)/(2 epsilon_(0))`
Now `E = (del varphi)/(del x) = (rho_(0))/(epsilon_(0))x - A = 0` for `x = 0`
if `A = (Delta varphi)/(d) + (rho_(0) d)/(2 epsilon_(0)) = 0`
then `rho_(0) = - (2 epsilon_(0) Delta varphi)/(d^(2))`
`E(d) = (rho_(0) d)/(epsilon_(0))`