## Dielectric Sphere In A Uniform Field

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# Dielectric Sphere in a Uniform Field

Consider an uncharged dielectric sphere of radius a and dielectric constant K_{1}, surrounded by a medium of dielectric constant K

_{2}and placed in a uniform electric field E0. The origin of the coordinate system is taken to be at the centre of the sphere and the X axis along E0. Let V

_{1}and V

_{2}be the potential function inside and potential functions inside and outside the sphere respectively. The solution to this problem must satisfy the following boundary conditions.

(i) ∇

^{2}V

_{1}= 0 and ∇

^{2}V

_{2}= 0, both inside and outside respectively, as the net charge on the sphere is zero.

(ii) V

_{1}must remain finite for all r

__<__a. V

_{2}must also remain finite at infinity.

(iii) V

_{1}= V

_{2}for r = a at all angles θ.

(iv) The normal component of

**D**must be continuous at r = a, i.e., D

_{1n}= D

_{2n}.

The potentials outside and inside the sphere can be written as:

V

_{2}= - E

_{0}r cos θ + (A/r

^{2}) cos θ … (1)

V

_{1}= Br cos θ + (C/r

^{2}) cos θ … (2)

C must be zero. Otherwise, the potential would become infinite at the origin (r = 0). Hence

V

_{1}= B

_{r}cos θ … (3)

According to the boundary condition (iii), V

_{1}= V

_{2}, for r = a

. : Ba cos θ = -E

_{0}a cosθ + (A/a

^{2}) cos θ or B = -E

_{0}+ a/a

^{3}… (4)

According to the boundary condition (iv),

D

_{1n}= D

_{2n}or K

_{1}ε

_{0}E

_{1n }= K

_{2}ε

_{0}E

_{2n}

Now, E

_{n}= -∂V/∂r

Hence, - K

_{1}(∂V

_{1}/∂r) r = a = -K

_{2}(∂V

_{2}/∂r)

_{r = a}

- K

_{1}B cos = -K

_{2}[-E

_{0}cosθ – (2A/a

^{3}) cos θ]

Or K

_{1}bB = -K

_{2}(E

_{0}+ 2A/a

^{3}) … (5)

Solving Eqns. (4) and (5), we get

A = (K

_{1}– K

_{2}/K

_{1}+ 2K

_{2}) E

_{0}a

^{3}and B = (-3K

_{2}/K1 + 2K

_{2})E

_{0}

Thus the potential function inside and outside the spheres are:

V

_{1}= - (3K

_{2}/K

_{1}+ 2K

_{2}) E

_{0}r cos θ … (6)

V

_{2}= - [1 – (K

_{1}– K

_{2}/K

_{1}+ 2K

_{2}) a

^{3}/r

^{3}] E

_{0}r cos θ … (7)

The electric field at any point inside the sphere is

E

_{1}= - ∂V

_{1}/∂x = (3K

_{2}/K

_{1}+ 2K

_{2}) E0 [. : = - (3K

_{2}/K

_{1}+ 2K

_{2})E

_{0}x] … (8)

The potential outside the sphere is equivalent to the applies field E0 plus the field of an electric dipole at centre of sphere with dipole moment.

p = 4πε

_{0}K

_{2}(K

_{1}– K

_{2}/K

_{1}+ 2K

_{2}) E

_{0}a

^{3}… (9)

oriented in the direction of applied field.

We know that when a dielectric sphere acquires a dipole moment in the field, it is polarized. The polarization is defined as the dipole moment per unit volume i.e.,

Polarization = P/(4/3)πa

^{3}= 3ε

_{0}K

_{2}(K

_{1}– K

_{2}/K

_{1}+ 2K

_{2}) E

_{0}… (10)

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