Electric potential: point charge in a hollow charged conductor

In summary, the electric potential as a function of radius for a charged conductor with a point charge is (3Q)/(8πε0r2).
  • #1
misa
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[solved] electric potential: point charge in a hollow charged conductor

Homework Statement


A hollow spherical conductor, carrying a net charge +Q, has inner radius r1 and outer radius r2 = 2r1. At the center of the sphere is a point charge +Q/2.

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d) Determine the potential as a function of r for 0 < r < r1.

Homework Equations



(π = pi)
For r > r2, the electric field is (3Q)/(8πε0r2).
For r1 < r < r2, the electric field is 0 (ie, field inside conductor is zero in static situations).
For 0 < r < r1, the electric field is Q/(8πε0r2).


The potential as a function of r for r > r2 (where voltage is taken to be 0 when r is infinite) is (3Q)/(8πε0r).

The potential as a function of r for r1 < r < r2 is (3Q)/(16πε0r1).


The Attempt at a Solution



My first instinct was to add the potential (3Q)/(16πε0r1) to Q/(8πε0r), which is the potential from infinity to r if the shell wasn't present. However, the answer is wrong. I also made many other fruitless attempts at this problem, but none of them very logical.

Can someone tell me what I'm doing wrong, and how to find this potential when r is within the cavity of the conductor?

I would greatly appreciate your help with this problem (and thank you in advance)!
 
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  • #2
When in doubt, go back to the definition of the potential at a point [itex]\vec{r}[/itex]:

[tex]V(\vec{r})=-\int_{\infty}^{\vec{r}}\vec{E}\cdot\vec{dl}[/tex]

The electric field has different values in different regions, so you break the integral into pieces:

[tex]V(\vec{r})=-\int_{\infty}^{\vec{r}}\vec{E}\cdot\vec{dl}=-\int_{\infty}^{r_2}\vec{E}_{\text{outside}}\cdot\vec{dl}+-\int_{r_2}^{r_1}\vec{E}_{\text{between}}\cdot\vec{dl}+-\int_{r_1}^{\vec{r}}\vec{E}_{\text{inside}}\cdot\vec{dl}[/tex]
 
  • #3
So are you saying that I should add all three potentials like this?
ie, (3Q)/(16πε0r1) + [Q/(8πε0)](1/r - 1/r1)

-I realize I derived the potential from r to r1 incorrectly.

-Also, I didn't include the potential in the region from r2 to infinity when I first solved the problem because I thought that (3Q)/(16πε0r1) basically came from the total potential from infinity to r1 (ie, potential is constant in that region aka the integral representing the change in potential is zero, so potential anywhere in r2 to r1 is just the potential as we approach r2). Am I right in assuming this or do I need to add (3Q)/(16πε0r) to the sum of the integrals too?
 
  • #4
Well, what does the middle integral actually add to the potential inside r<r1 ?
 
  • #5
No, it doesn't add to the integral because E is 0 in that region.

Okay, thank you for your help. :)
 

FAQ: Electric potential: point charge in a hollow charged conductor

What is electric potential?

Electric potential is a physical quantity that describes the amount of electric potential energy per unit charge at a given point in space. It is measured in volts (V) and is a scalar quantity.

How is electric potential calculated?

The electric potential at a point is calculated by dividing the electric potential energy by the charge at that point. Mathematically, it is expressed as V = U/Q, where V is the electric potential, U is the electric potential energy, and Q is the charge.

What is a point charge?

A point charge is a hypothetical charge that is concentrated at a single point in space. It is often used in theoretical calculations to simplify the analysis of electric fields and potentials.

What is a hollow charged conductor?

A hollow charged conductor is a conductor that has a cavity inside it and carries a net charge on its surface. The electric potential inside a hollow charged conductor is constant, and the electric field is zero.

How does a point charge in a hollow charged conductor affect the electric potential?

A point charge placed inside a hollow charged conductor will not affect the electric potential inside the conductor. This is because the electric field inside a conductor is zero, and the electric potential is constant. The point charge will only affect the electric potential outside the conductor.

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