(a) Potential at a point B defined as the amount of work done in bringing a unit positive charge from `oo` to that point.
Consider a charge `q_(0)` to brought from `oo` to point P.
Small amount of work done in moving the charge from A to B.
`dw=vecF.vecdx=q_(0)vecE.vecdx`
`=q.E.dxcos180^(@)=-q_(0)E.dx`
Total amount of work done to bring it from `oo` to P
`W=-intq_(0)E"dx=-int_(oo)^(r)q_(0)(1)/(4piin_(0))(q)/(x^(2))dx=-(qq_(0))/(4piin_(0)).underset(oo)overset(r)intx^(-2)dx`
`=-(qq_(0))/(4piin_(0))[-(1)/(x)]_(oo)^(r)=(qq_(0))/(4piin_(0))[(1)/(r)-(1)/(oo)]`
Now, `V=(W)/(q_(0))=(1)/(4piin_(0))(q)/(r)`.
(b) If charge + q is given to plate A, then charge -q is induced on the left face of plate B and charge +q on its right face. When plate B is earthed, the charge + q on its right face flows to earth. Due to charge The electric field between the two plates is related to the potential gradient as
`E=(dV)/(dr)` (in magnitude)
Between the two parallel plates, the electric field is uniform and perpendicular to the plates. Therefore, if V is potential difference between two plates, then
`E=(V)/(d)" ""for uniform field"((dV)/(dr)=(V)/(d))`
or V=Ed
If `sigma` is surface charge density of the plates, of the plates, then the electric field between the two plates is given by
`E=(sigma)/(epsi_(0))`
where `epsi_(0)` is absolute permittivity of the free space (If it assumed that medium between the plates is carcumm or air).
In the equation, substituting for E, we have
`V=(sigma)/(epsi_(0))d`
If A is the area of each plate, the
`sigma=(q)/(A)`
`:.V=(q)/(epsi_(0))(d)/(A)`
If C is the capacitance of the parallel plate capacitor, then
`C=(q)/(qd//epsi_(0)A)`
or `C=(epsi_(0)A)/(d)`.
