# electric potential at center of uniform electric field

it is said that the electric potential at the center of uniform electric field is zero. my question is that why is it zero? electric potential is the work done per unit charge.

$V = W/q$

and this work is continuously done on a positive test charge if it (charge) is placed in the electric field.

Consider two equal and opposite charges ($+q$ & $-q$) in space separated by a distance $2r$. An uniform electric field would exist between both acting from $+q$ to $-q$. The first thing is, Electric potential is a scalar quantity whereas Electric field is a vector..! In other words, Electric field is a measure of how the electric potential changes quickly with distance (gradient or the first derivative).

The electric potential at a distance $r$ from $+q$ would be $V_1=\frac{kq}r$

Now, the electric potential at a distance $r$ from $-q$ is $V_2=-\frac{kq}{r}$

The net (effective) potential at midpoint ($r$) is $V=V_1+V_2=0$

In case of Electric field, it is non-zero. Because, we would specify the direction only...

Regarding your case, A test (point) charge not necessarily positive. It's just to indicate the existence of an electric field. In presence of a charge, the test charge would experience a force. That's all :-)

• The field between two point charges is not uniform. So your discussion is not relevant for the OP. – nasu Oct 7 '18 at 13:54

The only potential differences matter in Electromagnetism.

So in the case of a uniform electric field along the x direction produced by 2 plates. With the negative one being left and positive one being right. Now I will assume your plates are infinitely long in the Y,Z plane.

In the region in between the plates. $\phi = - k x +c$.

$E = -\nabla \phi = k$

Here $\phi$ is the electric potential. K is the constant electric field, and c is any arbitrary constant.

Now coming to your question. You say that electric potential at the center of a uniform potential is constant. Now in principle one can shift the potential by any constant. and it would still be alright. It is conventionally taken to be zero at infinity.

If you impose this convention, The potential outside the plates is zero. There is sharp jump in the Potential at the Plates to account for sudden change in charge. This jump is negetive on the positive plate and positive for the Negetive plate. Since the charges are equal The jumps must be equal in magnitude. The electric potential will begin to fall as you move towards the positive plate linearly. The symmetry of the situation allows us to establish that he mid-point will have zero potential.

The location of the zero for potential (also known as reference or ground) is entirely arbitrary. You have to arbitrarily choose some point, so you are free to choose the center if you like. If you don’t like that choice then you are completely free to choose some other point. The physics does not depend on this choice. As long as you are consistent, you will still get the right answers.