0

I read in a few books that there is always an electric field if there is an electric potential. I went through this but the question on this page only states that there is an electric field and it is specific only to batteries. It may be very obvious, but I don't get the exact logic. I can understand that there is electric potential if there is a field, but the opposite doesn't seem so obvious to me. Why, if there is a potential difference, is an electric field generated?

Community
  • 1
Dieblitzen
  • 1,637
  • 2
  • 14
  • 16
  • 7
    The potential is defined as the thing whose derivative is the field. – fqq Jun 01 '16 at 10:36
  • 1
    An electric potential has the same meaning as a mechanical potential: it defines a specific amount of work that the field can do on a unit charge that is being moved between any two points. If there is no field, there is no force, hence no work is being performed, hence there is no need to talk about a potential. Now, even if there is a field there does not have to be a potential because the field does not have to be conservative. – CuriousOne Jun 01 '16 at 10:37
  • 1
    It's also useful to remember, that the potential is merely a mathematical construct invented by people. Only fields are real. So if there's a potential difference, you have to ask not "why is there a field then?"; rather "why is there a potential difference?" and the answer is "evidently, due to a field". – LLlAMnYP Jun 01 '16 at 12:16
  • 2
    Why do there have to be rabbits if there is a rabbit population? (Answer: because the population measures something about the rabbits.) – WillO Jun 01 '16 at 12:49

2 Answers2

3

Start by noting that the electrical potential is an energy per unit charge.

In an electric field $E$ the field produces a force on a charge $Q$ of:

$$ F=EQ $$

so if we move the charge a distance $dr$ the work done is just force times distance or:

$$ W=EQ\,dr $$

The work done per unit charge is $E\,dr$, and this is what we mean by the change in the potential:

$$ dV = E\,dr \tag{1} $$

Now, you start by saying you understand why there must be a potential when there is an electric field, and obviously it's because if there is a field then there is a force on a charge, and there must be an associated energy change given by equation (1) when we move that charge. We get the energy change, i.e. the potential difference, by integrating equation (1):

$$ \Delta V = \int E\,dr \tag{2} $$

But we can rearrange equation (1) to give:

$$ E = \frac{dV}{dr} \tag{3} $$

This is telling us that if there is an energy change $dV$ when we move a unit charge a distance $dr$ then there must have been some force acting on the charge to do that amount of work. This force is the field $E$ times the (unit) charge, so the conclusion is that if the potential changes with distance there has to be a field $E$ present.

John Rennie
  • 355,118
1

The mathematical definition that John Rennie gave explains it well, but I'd like to give an intuitive answer to your question. Imagine a rubber sheet horizontally stretched. The height of the rubber sheet at a point is equivalent to the electric potential at that point. Now, since as of now there is no disturbance at all, the height (potential) throughout the rubber sheet will be equal, which we can obviously take to be 0. Now, assume that there is some extra potential which has accumulated at some point on the rubber sheet. Now, since potential at a point is proportional to the height of the rubber sheet at that point, you can expect a bulge in the fabric. It'll look as if someone has poked a finger from the bottom of the rubber sheet. If there is a potential difference at some point (an irregularity), something has ought to have caused it right? The only thing that can cause an irregularity is a charge. And each and every charge exerts a force. (Some force has to be applied to cause the bulge in the rubber sheet). Therefore, the potential difference has to be caused by the force exerted by some charge as we need an imbalance in the potential at a specific point on the rubber sheet. And we already know that where there is a charge, there is a field.

Krish Wadhwana
  • 123