# The electric potential inside a conducting sphere with charge Q

• j04015
j04015
Homework Statement
The electric potential inside a conducting sphere with charge Q is

A. Zero
B. Nonzero
Relevant Equations
F=MG
F=EQ
If there is no field inside the conductor, how can there be electric potential?

I think of potential very similar to gravity, as how much energy would be required to move a particle of mass/charge against the gravitational/electric field.

If there is no field at all, how would there still be electric potential in the sphere?

The field being zero means that the gradient of the potential is zero, not that the potential is zero.

MatinSAR
Orodruin said:
The field being zero means that the gradient of the potential is zero, not that the potential is zero.
Can you explain that in more detail?

j04015 said:
Can you explain that in more detail?
The gradient tells you how rapidly the potential changes, right? And also in which direction it changes most rapidly.

Within the interior of a [uniformly] charged sphere the potential is unchanging. The potential at any interior point is the same as everywhere else in the interior. So, in the interior, the gradient of the potential is the zero vector.

But that says nothing about the exterior. Or about the potential of an interior point relative to a point at infinity.

To use an analogy, if the bottom of a valley (or the top of a mesa) is flat, that says nothing about the altitude there.

MatinSAR
This question could have been formulated better. One does not ask if the electric potential is zero or non-zero inside the sphere without specifying where the potential is taken to be zero. If not specified and the zero of potential were left to the reader, then either answer could be correct depending on the reader's choice of reference.

In its current form this question is like asking whether a person at rest on the x-axis is on the origin or away from the origin. To @j04015 : Your question
j04015 said:
If there is no field at all, how would there still be electric potential in the sphere?
is equivalent to asking, "if the person is not moving, how could he be away from the origin?" Do you see the analogy?

jbriggs444 and MatinSAR
kuruman said:
One does not ask if the electric potential is zero or non-zero inside the sphere without specifying where the potential is taken to be zero
It could have been specified elsewhere that the potential is taken to be zero at infinity. That is likely the convention used by the author.

j04015 said:
Homework Statement: The electric potential inside a conducting sphere with charge Q is

A. Zero
B. Nonzero
Relevant Equations: F=MG
F=EQ

If there is no field inside the conductor, how can there be electric potential?
If the potential is uniform (same everywhere) then the field is zero.

Good Query! Inside a conducting sphere with charge Q, the electric potential is indeed nonzero. Here’s why: In electrostatics, the electric field inside a perfect conductor in equilibrium is zero. However, the electric potential remains constant throughout the conductor. This means that while there is no field (i.e., no force acting on a charge within the sphere), the potential is the same everywhere inside the sphere. Think of it as a plateau—no slope (field), but at a certain height (potential). So, the correct answer is B. Nonzero.

rayansmith said:
Good Query! Inside a conducting sphere with charge Q, the electric potential is indeed nonzero. Here’s why: In electrostatics, the electric field inside a perfect conductor in equilibrium is zero. However, the electric potential remains constant throughout the conductor. This means that while there is no field (i.e., no force acting on a charge within the sphere), the potential is the same everywhere inside the sphere. Think of it as a plateau—no slope (field), but at a certain height (potential). So, the correct answer is B. Nonzero.
As noted in posts #5 and #6, it does not follow that the potential inside is nonzero.
It will be nonzero if (a) we adopt the common, but not universal, convention that the potential at infinity is zero and (b) there are no other charges around worthy of note.
For example, suppose the sphere is inside a larger sphere of equal and opposite charge.

rayansmith
haruspex said:
(a) we adopt the common, but not universal, convention that the potential at infinity is zero
In this context, it should be noted that some charge distributions simply don’t allow the potential at infinity to be set to zero.

Orodruin said:
In this context, it should be noted that some charge distributions simply don’t allow the potential at infinity to be set to zero.
Right, like an infinite uniformly charged sheet.

haruspex said:
As noted in posts #5 and #6, it does not follow that the potential inside is nonzero.
FWIW it doesn't necessarily imply that the potential is zero simply because the field is zero. I realize this distinction was not made clear in the original question. It's a poorly-worded question.

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