# Voltage and Electric fields

it says "The electric field inside a conducting sphere is zero, so the potential remains constant at the value it reaches at the surface:"

if the electric field inside the sphere is 0, then wouldn't that mean the rate of change of voltage against change in distance is also 0? Would that then suggest, within the sphere we get maximum voltage?

I'm having a hard time understanding why the potential inside = potential at the surface

Thanks!

## Answers and Replies

it says "The electric field inside a conducting sphere is zero, so the potential remains constant at the value it reaches at the surface:"

if the electric field inside the sphere is 0, then wouldn't that mean the rate of change of voltage against change in distance is also 0?
Yes. The definition of the electric field is the derivative of the voltage (i.e., the potential.) Mathematically, $$\vec{E}(\vec{x})=-\nabla\phi(\vec{x})$$ where $\vec{E}(\vec{x})$ is the electric field at the point $\vec{x}$, and $\phi(\vec{x})$ is the voltage at the point $\vec{x}$. The symbol $\nabla$ represents the gradient operator, which is a generalization of a one-dimensional derivative.
Would that then suggest, within the sphere we get maximum voltage?
Not necessarily. This would be analogous to the question:
"Just because you are standing on level ground within some area, would that suggest you are standing on the top of a mountain?"
Just because the potential isn't changing does not imply it is a global minimum or maximum.

I'm having a hard time understanding why the potential inside = potential at the surface
The charges on the conductor, interacting with the external field and with the fields of the other charges within the conductor, move freely throughout the conductor in response to the net field. The charges move until they find a configuration at which no net field acts on them [otherwise they would continue to move!].

On the surface of a conductor, you may have some sort of non-differentiable point in the potential, so at the surface you might be tempted to think that the charges should be escaping the surface of the conductor, but in reality their confinement on the surface is due to internal interactions with the charges comprising the conductor, and the potential is indeed smooth. But in classical EM when you consider a 3D conductor with a perfectly thin 2D surface, you typically impose continuity as the obviously mathematical choice for how the potential changes at the surface (ohterwise, discontinuities in the potential would imply infinite forces.)

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You can say that the magnitude of electric potential inside the sphere is maximum only if the sphere was completely isolated and there are no other electric field lines or charges around it.