Question

Q2a) Show that the potential at the surface of a uniformly charged sphere of radius R is V=Kq/R where K = 1/4πɛ₀ and the zero of potential is at infinity). Hence derive an expression for the potential energy of the sphere. (Hint, assemble the sphere from the charges at infinity)

b) A parallel plate capacitor has conducting plates, each of area A, situated at y=0 and y=d. two parallel dielectric sheets occupy the space between the plates. Dielectric of relative permittivity εᵣ occupies the region y=0 to y=d, and the remaining space, from d to d contains having relative permittivity εᵣ₂.

i) calculate from first principles, the capacitance of this device
ii) calculate the surface polarization charge at the boundary between the dielectrics when free charge +q and -q is put on the lower and upper plates, respectively.

Answer 1

2 (a) Consider a uniformally charged sphere of Radious "R" having charge q

Electric Feild at point ' p ' at r form center consider quession surface of radius r

According to gaurs Law

Now potential

at surface r = R

Since charge q is uniformally distributed on sphere, let f be the charge per unit volume.

The phere may be supported to be formed by a spherical shells.

Consider are shell of radious r and thickmen dx

Volume of sphere of

Radious x is

Charge on sphere of radious x is

Volume of spherical shell

Charge on spherical shell

Electro static potential energy b/w solid spherical core and spherical shell

Total electro static

Now Arrangement is equal to these two caparter with area A and thickness d in series so their Resultant