Suppose the charge density rho of a solid nonconducting sphe

Suppose the charge density, rho, of a solid non-conducting sphere. radius=R. is given by rho = beta r^3 ; (beta is a Constant. a. Find beta as a function of the total charge Q on the sphere and the sphere radius R.b. Find the electric field as a function of, Q, R and the distance, r, from the center inside the sphere,ie r

Solution

Here, the sphere is non-conducting in nature and has a charge density of p = r^3. We would need to determine the toal charge in terms of and then using Q, we can find the expression for . Also, we will use Gauss\' law so as to determine the electric field at any point inside the sphere.

Part a.) Let us assume a thin spherical shell at radius r, thickness dr and concentric with the given sphere. The density at this distance r from the centre would be r^3 also, the volume of this shell would be 4r^2 dr

Therefore the charge contained in this spherical shell would be: 4 r^5 dr

That is, dq = 4 r^5 dr

we can integrate the above expression from 0 to R and find the total charge Q, as

Q = dq =  4 r^5 dr = 4 R⁶/6

Therefore, = 6Q/4R⁶ is the required expression.

Part b.) Now, let us consider a Gaussian surface of radius r and concentric with the given sphere.

By Gauss\' law, we have: Eds = Q(enclosed) / o

or, E(4r²) = 4 r⁶/6o

or E = 4 r⁴/6o

We can replace with the expression above to get:

E = (Q/R⁶)(r⁴/o) = Qr⁴ / R²o which is the required expression for electric field at a point inside the sphere.