# V(center) of charged metal sphere inside a grounded shell

• Pushoam
In summary, if you take another sphere of charge -q of radius a with uniform charge density, then the potential on the spherical region from the radius a to b is the same to that of the original question.
Pushoam

## The Attempt at a Solution

Potential at the center of conducting charged sphere surrounded by a grounded shell

If I take another sphere of charge –q of radius a with uniform charge density, then the potential on the spherical region from the radius a to b is same to that of the original question.

According to Uniqueness theorem, the V at the center of the sphere due to the above system should be equal to that of the original question.

Since the conducting sphere is an equipotential, the potential due to this sphere at center is same to that of the surface = ## \frac { q}{4 \pi \epsilon _0 R} ## .The potential due to the image sphere of charge –q at the center is ## \frac { -q}{4 \pi \epsilon _0 a} ## .

So, the total potential at the center is ## \frac { q}{4 \pi \epsilon _0 R} ## - ## \frac { q}{4 \pi \epsilon _0 a} ## , option (d).

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Looks OK, but you don't need to invoke the Uniqueness Theorem and image charges. Note that grounding the sphere effectively brings the reference point from infinity to ##r=a##. Then just use Gauss's Law to find the E-field in the region ##R < r < a## and then use ##V(R)-V(a)=-\int_a^R{E~dr}##.

kuruman said:
Note that grounding the sphere effectively brings the reference point from infinity to r=a.
I didn't note this.
I was in the habit of taking reference point at infinity, so I thought of solving it that way even when an easier approach was near.

Thanks for pointing it out.

Pushoam said:
didn't note this.
If you are prepared to trust the list of options then, noting that, you can get to the answer very quickly. It means that the radius b cannot be relevant, so we rule out options a and c.
As radius a tends to R, the induced charge cancels the charge on the inner sphere. This leaves d as the only option.

Pushoam

## 1. What is the purpose of a charged metal sphere inside a grounded shell?

The purpose of a charged metal sphere inside a grounded shell is to create a uniform electric field within the shell. This is because the charges on the surface of the sphere will distribute themselves evenly on the inner surface of the shell, resulting in a uniform charge density.

## 2. How does the location of the charged metal sphere affect the electric field inside the grounded shell?

The location of the charged metal sphere inside the grounded shell does not affect the electric field inside the shell. This is because the electric field inside a conductor is always zero, and since the shell is grounded, the charges on the inner surface of the shell will rearrange themselves to cancel out the electric field of the charged sphere.

## 3. What happens to the electric potential at the center of the charged metal sphere inside a grounded shell?

The electric potential at the center of the charged metal sphere inside a grounded shell is zero. This is because the potential at any point inside a conductor is equal to the potential at the surface, and since the potential at the surface of the sphere is zero (due to it being grounded), the potential at the center is also zero.

## 4. Can a charged metal sphere inside a grounded shell experience a net force?

No, a charged metal sphere inside a grounded shell cannot experience a net force. This is because the charges on the inner surface of the shell will distribute themselves evenly, resulting in a cancellation of any external electric field. Therefore, the net force on the charged sphere will be zero.

## 5. How does the charge of the metal sphere affect the electric field inside the grounded shell?

The charge of the metal sphere does not affect the electric field inside the grounded shell. This is because the electric field inside a conductor is independent of the charge on the conductor. The only factor that affects the electric field inside a grounded shell is the presence and location of external charges or conductors.

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