# Electric Fields potential of zero

1. Jan 8, 2009

### nokia8650

Can somone please explain why the following is not true:

"Electric potential is zero whenever the electric field strength is zero"

I know that the Field strength is the potential gradient, however why is the above true?

2. Jan 8, 2009

### gabbagabbahey

Well, the gradient of any constant is always zero isn't it? So why couldn't the Potential be any constant? Certainly zero is one such constant, but there are infinitely many others.

3. Jan 8, 2009

### nokia8650

Thanks for the reply. Would it be possible for you explain it in a more of a physical context, as I understand that is the case mathematically, however not in a physical context.

Thank you very much

4. Jan 8, 2009

### turin

There are two ways to reason this. One is to realize that there is always an arbitrary integration constant (an example of an electromagnetic gauge choice). The other is to ask yourself, "what is the electric field corresponding to a unifrom potential of, say, 100 V?" and then consider the logical corrolary.

5. Jan 8, 2009

### gabbagabbahey

Well, to calculate the electrostatic potential is defined only up to some constant. This is due to the definition: $\vec{E}=-\vec{\nabla}V$. Physically, that means that you can never measure the potential at a point; only the difference in potential between two points can be measured. Typically, when physicists speak of the potental at a point, they mean 'the potential at a point relative to some agreed upon reference point'. Often one takes the reference point to be infinity (i.e. Very very far away from the field point you are interested in) and defines the potential to be zero there. But, you can define the refernence point to be anywhere you like, and the reference potential to be any value you like; so it is true that you can always define the potential to be zero at some point of interest, but that affects the potential everywhere else which is measured relative to that point. So whether or not the potential is zero wherever E=0 depends on where you have set your reference point.

6. Jan 8, 2009

### Defennder

A counter-example to disprove that claim may be given as follows:
Consider the electric field at the centre due to a charged spherical shell. It is 0. What is the electric potential at the centre?

7. Jan 8, 2009

### nokia8650

Im not sure how I'd calculate the electrical potential, however there would be some as work would have be done to move a charged particle from there to infinity.

Thank you to all of the above posters for the help.

8. Jan 8, 2009

### Defennder

In that case of the charged spherical shell you only need to do a volume integral over the spherical volume for each electric potential contribution due to dV:
So we have $$dW = \frac{p_v dV}{4\pi \varepsilon_0 r}$$. Integrate over the volume. A simpler example would be the centre of a charged ring. It also has 0 net electric field strength but non-zero electric potential there.

9. Jan 9, 2009

### gabbagabbahey

Of course, your method implicitly assumes that the reference point has been set to infinity.