A sphere of radius R carries a polarization

P ( r ) = kr,

where k is a constant and r is the vector from the center.

(a) Calculate the bound charges σb and ρb.

(b) Find the field inside and outside the sphere.

Solution 10P

(a)

Step 1 of 8</p>

To find the bound charges and for a sphere of radius R carrying polarization

Given by , where k is a constant and r is vector from the center

Step 2 of 8</p>

For the bound surface charge, the unit vector on the surface of the sphere is equal to the radial unit vector . So the bound surface charge is given by,

Therefore, …..(1)

Step 3 of 8 Now, the bound volume charge is given by,

= …..(2)

And the equation (2) can be written as,

Therefore, …..(3)

(b)

Step 4 of 8</p>

Here, we first have to consider the region outside the sphere. And the electric field in this region due to the surface charge is equal to,

Therefore, substituting the value of in the above equation we get,