Find Electric Field Around a Non-Conducting Sphere

[tuggler]

Homework Statement

I. A non-conducting sphere of radius a has a spherically symmetric, but non-uniform charge distribution is placed on it, given by the volume density function: p(r) = C·r, where C is a positive constant, and 0 < r < a.

a. Find an algebraic expression for the total charge Q on the sphere, in terms of the parameters C and a.

II. Use Gauss' Law to find an algebraic expression for the magnitude of the electric field at a distance R from the origin, in each of the following regions. Express your answer in terms of the following four parameters: the electrostatic constant k; the radius a of the sphere; the total charge Q on the sphere; and the radial distance R from the origin to the field point.

b. Within the insulating sphere (i.e. for R <a):

c. Outside the sphere (i.e. for R > a):

The Attempt at a Solution

I got Q = Cpi(a^4) for part a which is correct.


For part 2 I am stuck both b and c.

I got [(KQR^4)/((a^4)(r^2))] which is incorrect.

 

[ehild]

For part 2 I am stuck both b and c.

I got [(KQR^4)/((a^4)(r^2))] which is incorrect.

Is this your answer for b) or c)? What is r in your formula?

ehild

 

[tuggler]

r is what I got with determining K. I got $$\frac{1}{4\pi \epsilon_0 r^2},$$ but I replaced the 1/4pi e_0 with K.

 

[vela]

What does $r$ represent physically? $R$ is the distance from the center of the sphere; $a$ is the radius of the sphere. What is $r$?

You need to show your work. Just posting an incorrect answer is next to useless for us to see where you're getting stuck.

 

[Nickie]

I would first like to commend the student for their correct expression for the total charge Q in part a. This shows a good understanding of the problem and the given parameters.

Moving on to part b and c, the student's attempt at using Gauss' Law is commendable, however, there are a few errors in the expression. First, the electrostatic constant k should be squared in the numerator, and the radial distance R should also be squared in the denominator. Additionally, the total charge Q should be multiplied by the volume density function p(r) to account for the non-uniform charge distribution.

Therefore, the correct expressions for the electric field in each region would be:

b. Within the insulating sphere (i.e. for R < a): E = (kQp(r)R^2)/(2a^3)

c. Outside the sphere (i.e. for R > a): E = (kQa^3)/(R^2)

It is important to note that in the case of a non-conducting sphere, the electric field will not be constant throughout the entire region, as it would be for a conducting sphere. This is due to the non-uniform charge distribution on the sphere.

In conclusion, the student has shown a good understanding of the problem and has made a good attempt at using Gauss' Law to find the electric field. However, there were a few errors in the expression that needed to be corrected. As a scientist, it is important to carefully check and double-check our calculations to ensure accuracy.