Find Electric Field Around a Non-Conducting Sphere

In summary, the conversation discusses a non-conducting sphere with a non-uniform charge distribution and the use of Gauss' Law to find the electric field at different distances from the origin. An algebraic expression for the total charge on the sphere is given, and the question asks for the electric field within and outside the sphere. The attempted solution for part a is correct, but the solution for part 2 is incorrect and further clarification is needed.
  • #1
tuggler
45
0

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.
 
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  • #2
tuggler said:
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
 
  • #3
r is what I got with determining K. I got [tex]\frac{1}{4\pi \epsilon_0 r^2},[/tex] but I replaced the 1/4pi e_0 with K.
 
  • #4
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.
 
  • #5



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.
 

1. What is the formula for electric field around a non-conducting sphere?

The formula for electric field around a non-conducting sphere is given by E = kQr/ε0r3, where E is the electric field, Q is the charge of the sphere, r is the distance from the center of the sphere, k is the Coulomb's constant, and ε0 is the permittivity of free space.

2. How do I calculate the electric field at a certain point near the sphere?

To calculate the electric field at a certain point near the sphere, you will need to know the distance from the center of the sphere to the point, as well as the charge of the sphere. Using the formula E = kQr/ε0r3, you can plug in these values to calculate the electric field at that point.

3. Does the electric field change if the sphere is charged or uncharged?

Yes, the electric field around a non-conducting sphere will be affected by the charge of the sphere. The larger the charge of the sphere, the stronger the electric field will be at a given distance from the center of the sphere.

4. How does the electric field around a non-conducting sphere differ from that around a conducting sphere?

The main difference between the electric field around a non-conducting sphere and a conducting sphere is that the electric field inside a conducting sphere is zero, while the electric field inside a non-conducting sphere is not necessarily zero. This is because charges are able to move freely within a conducting sphere, while they are fixed in place in a non-conducting sphere.

5. Can the electric field around a non-conducting sphere be negative?

Yes, the electric field around a non-conducting sphere can be negative. This simply means that the direction of the electric field at a certain point is opposite to the direction of the electric field at another point. The magnitude of the electric field will still be positive, as it is determined by the charge of the sphere and the distance from the center of the sphere.

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