What Are the Electric Field and Potential in a Non-Conducting Hollow Sphere?

[NYK]

Homework Statement

https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xaf1/v/t1.0-9/10155063_1485929725016579_7557468199677155885_n.jpg?oh=d575aa48176de7ecda27201b7ce35a5b&oe=54F7F368&__gda__=1425195231_44abd00cc231109df2fbe8d44cef9869

Homework Equations

∫E⋅da⋅n = 4πkQ_enc
ΔV(voltage) = -∫Edr

The Attempt at a Solution

To be honest I am having trouble starting the problem and defining the Q_enc for the three areas of the sphere.

I do believe that for r < R ⇒ E(r) = 2QK/r²

and for r > R ⇒ E(r) = KQ/r²

I know that the answer is listed for R<r<2R but i can't seem to come up with that answer.

I think I am just having problems with defining the Q_enc

for R<r<2R does Q_enc = 2Q_{pt charge} - Q_induced = Q sound right?

 

[vela]

Homework Statement

https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xaf1/v/t1.0-9/10155063_1485929725016579_7557468199677155885_n.jpg?oh=d575aa48176de7ecda27201b7ce35a5b&oe=54F7F368&__gda__=1425195231_44abd00cc231109df2fbe8d44cef9869

Homework Equations

∫E⋅da⋅n = 4πkQ_enc
ΔV(voltage) = -∫Edr

The Attempt at a Solution

To be honest I am having trouble starting the problem and defining the Q_enc for the three areas of the sphere.

I do believe that for r < R ⇒ E(r) = 2QK/r²

and for r > R ⇒ E(r) = KQ/r²

That's right.

I know that the answer is listed for R<r<2R but i can't seem to come up with that answer.

I think I'm just having problems with defining the Q_enc

for R<r<2R. Does Q_enc = 2Q_{pt charge} - Q_induced = Q sound right?

No. The ball is non-conducting, so there won't be any induced charges. You have a charge -Q spread out uniformly within the ball. When R < r < 2R, you have to figure out what fraction of the ball is enclosed inside of the sphere of radius r. It's a geometry problem.

 

[NYK]

That's right.No. The ball is non-conducting, so there won't be any induced charges. You have a charge -Q spread out uniformly within the ball. When R < r < 2R, you have to figure out what fraction of the ball is enclosed inside of the sphere of radius r. It's a geometry problem.

I understand that, so if it was a conducting sphere the charge would be Q_enc = Q_{pt charge} +λ(π(r²-R²))?

 

[vela]

No. How did you come up with that expression?

 

[HalfLostAndSearching]

I would approach this problem by first defining the properties of a non-conducting hollow ball. This means that the material of the ball does not allow for the flow of electric current, and therefore, any charge placed on it will remain in place.

Next, I would consider the electric field inside and outside the ball. Since the ball is hollow, there is no charge within the cavity and therefore, by Gauss's Law, the electric field inside the ball is zero. Outside the ball, the electric field can be calculated using the given equations for r < R and r > R.

To determine the potential difference within the ball, I would integrate the electric field from the surface of the ball (r=R) to the center of the ball (r=0). This will give the potential difference between the two points.

As for defining the Qenc, it would depend on the situation. If there is a point charge (Q) placed at the center of the ball, then Qenc would be equal to Q. If there is no point charge, then Qenc would be zero.

For the given problem, it seems that the answer is for a point charge placed at the center of the ball. In this case, Qenc would indeed be equal to 2Qpt charge - Qinduced. However, if the charge was not placed at the center, the answer would be different.

Overall, it is important to carefully consider the given conditions and use appropriate equations to solve the problem.