Nonconducting spherical shell with uniform charge

In summary: Your patience and guidance helped me understand the problem better. I appreciate it!In summary, the electric field inside a nonconducting sphere with a spherical cavity of radius r1 and charge Q distributed uniformly in the "shell," between r = r1 and r = r0, can be determined by using Gauss's Law and a Gaussian surface of radius r. The charge enclosed within this surface is qenc = [Q(r^3 - r1^3)] / (r0^3 - r1^3), and the electric field is given by E = Q(r^3 - r1^3) / [4πr^2ε0(r0^3 - r1^3)].
  • #1
ooohffff
74
1

Homework Statement


Suppose the nonconducting sphere of Example 22-4 has a spherical cavity of radius r1 centered at the sphere's center (see the figure). Assuming the charge Q is distributed uniformly in the "shell" (between r = r1 and r = r0), determine the electric field as a function of r for the following conditions.

r1 < r < r0

22-31.gif


Homework Equations


Gauss's law

The Attempt at a Solution



I know the electric field is 0 in the inner cavity. The electric field outside of the shell is Q/(4pi*E0*r^2). How would I derive the equation for the electric field inside the shell? Charge per unit volume should be p=Q/(4/3*pi(r0-r1)^3) right?
 
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  • #2
ooohffff said:

Homework Statement


Suppose the nonconducting sphere of Example 22-4 has a spherical cavity of radius r1 centered at the sphere's center (see the figure). Assuming the charge Q is distributed uniformly in the "shell" (between r = r1 and r = r0), determine the electric field as a function of r for the following conditions.

r1 < r < r0

22-31.gif


Homework Equations


Gauss's law

The Attempt at a Solution



I know the electric field is 0 in the inner cavity. The electric field outside of the shell is Q/(4pi*E0*r^2). How would I derive the equation for the electric field inside the shell? Charge per unit volume should be p=Q/(4/3*pi(r0-r1)^3) right?
That charge density is incorrect.

A3 - B3 ≠ (A - B)3 .
 
  • #3
SammyS said:
That charge density is incorrect.

A3 - B3 ≠ (A - B)3 .

Mmm yes, your'e right it should be Q/(4pi/3(r0^3-r1^3)). But where do you go from there?
 
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  • #4
ooohffff said:
Mmm yes, your'e right it should be Q/(4pi/3(r0^3-r1^3)). But where do you go from there?
How much charge there interior to a sphere of radius, r, where r1 < r < r0 .
 
  • #5
SammyS said:
How much charge there interior to a sphere of radius, r, where r1 < r < r0 .

Could you rephrase that and be more specific?
 
  • #6
Would it be the same as the electric field outside the sphere?
 
  • #7
ooohffff said:
Would it be the same as the electric field outside the sphere?
No, electric charge is not equal to electric field.

You want to find what the electric field is interior to the "shell". That's where r1 < r < r0 .

You mention Gauss's Law.
 
  • #8
SammyS said:
No, electric charge is not equal to electric field.

You want to find what the electric field is interior to the "shell". That's where r1 < r < r0 .

You mention Gauss's Law.

Right its

∫E⋅da=qenc0

So I keep getting E4πr^2=qenc0

so that would mean E would equal the same as the electric field as the outside of the sphere, which I know is wrong. What am I doing wrong?
 
  • #9
ooohffff said:
Right its

∫E⋅da=qenc0

So I keep getting E4πr^2=qenc0

so that would mean E would equal the same as the electric field as the outside of the sphere, which I know is wrong. What am I doing wrong?
For r between r1 and r0, what do use for your Gaussian surface?
 
  • #10
SammyS said:
For r between r1 and r0, what do use for your Gaussian surface?

Would it be r1^2 instead of r^2?
 
  • #11
ooohffff said:
Would it be r1^2 instead of r^2?
No.

If you want to know the electric field at a distance r from the origin, use s sphere of radius r.

How much electric charge is enclosed within this sphere?

In the case of your problem the only charge inside this sphere resides in the spherical shell with inner radius r1 and outer radius r. Right ?

So, how much charge is that ?
 
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  • #12
So if charge density =

Q/ (4π/3 ( r03-r13)) = qenc/(4π/3 (r3-r13)).

Then,

Q(r3-r13) = qenc (r03-r13)

qenc = [Q(r3-r13)] / (r03-r13)

E4πr2 = Q(r3-r13)] / ε0(r03-r13)

E = Q(r3-r13)] / 4πr2ε0(r03-r13)
 
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  • #13
Can someone verify this?
 
  • #14
ooohffff said:
Can someone verify this?
Patience.

ooohffff said:
So if charge density =

Q/ (4π/3 ( r03-r13)) = qenc/(4π/3 (r3-r13)).

Then,

Q(r3-r13) = qenc (r03-r13)

qenc = [Q(r3-r13)] / (r03-r13)

E4πr2 = Q(r3-r13)] / ε0(r03-r13)

E = Q(r3-r13)] / [4πr2ε0(r03-r13) ]

That's the right idea. To be perfectly correct, the entire denominator should be enclosed in parentheses, else use some other grouping symbol.
 
  • #15
SammyS said:
Patience.
That's the right idea. To be perfectly correct, the entire denominator should be enclosed in parentheses, else use some other grouping symbol.

Haha, thank you SammyS!
 

FAQ: Nonconducting spherical shell with uniform charge

1. What is a nonconducting spherical shell with uniform charge?

A nonconducting spherical shell with uniform charge is an object that has a uniform distribution of electric charge on its surface, but does not allow the flow of electric current. This means that the charges on the surface of the shell do not move, but they still exert electric forces on other charged objects.

2. How is the electric field inside a nonconducting spherical shell with uniform charge?

The electric field inside a nonconducting spherical shell with uniform charge is zero. This is because the electric charges on the surface are evenly distributed and cancel each other out, resulting in a net electric field of zero inside the shell.

3. What is the electric potential of a nonconducting spherical shell with uniform charge?

The electric potential of a nonconducting spherical shell with uniform charge is constant at every point on the surface of the shell. This is because the electric potential is directly proportional to the electric field, which is zero inside the shell and constant on the surface.

4. How does the electric field change if the charge on the nonconducting spherical shell is increased?

The electric field inside the shell remains zero, but the magnitude of the electric field on the surface of the shell increases. This is because the total charge on the surface increases, resulting in a stronger electric force.

5. Can a nonconducting spherical shell with uniform charge be used to shield from external electric fields?

Yes, a nonconducting spherical shell with uniform charge can act as a shield from external electric fields. This is because the electric field inside the shell is zero, so any external electric field will not affect the charges on the surface and will be cancelled out.

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