# Electric Potential, V

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## Electric Potential, V

We've discussed **potential** in the section on current electricity. We said that it was measured in volts. That's confirmed here.

**Electric potential is the electrical potential energy per unit charge (ie. per coulomb) at a point in a field.**

It is a **scalar.**

It only depends on the charge causing the field - it is a property of the field!

**An equation for it is:**

* where*.

**W** = work done moving Q through the field (in J).

and

**Q** = the charge being moved through the field (in C)

* Example:* Just as in gravitational fields (see those notes for more detail), there is some confusion about where the value of zero potential is. It is in fact at infinity when charges have no effect on each other. If they are not attracting or repelling each other they can't be expected to do any work on each other!

So imagine bringing a positive charge from infinity (zero potential) in towards another positive charge. You are going to have to push it (do work on it) so it gains E_{p}. That's easy! Maximum potential close to the object, zero potential at infinity.

Now think about bringing a negative charge from infinity towards the positive charge. They attract! So at infinity they have zero potential (as always) but, as they get closer together the E_{p} reduces (i.e. becomes less than zero). Right next to each other, they have very little E_{p} - if you let go of them, they just stay there! So, as with gravitational fields, you can have **negative values of potential.**

**A detailed definition for potential is:**

*''The potential at a point in a field is equal to the work done per coulomb in moving a positively charged particle from infinity to that point in the field.''*

* Note:* it's a

**positive charge**we're moving.

The equation for **potential**, V is:

**where**

Q = the charge causing the field

ε = the permittivity, usually ε_{0}.

r = the separation between the centre of the charge and the point you are interested in.

**Example 1**

**What's the potential at a point 2m from a +3C charge?**

**Answer 1**

You need to consider whether it is going to be a positive or negative value. The definition of potential says it's the work done moving a **positive** charge into the field. Clearly, you would have to **push** a positive charge into this field against its will. You are going to have to do work on it. It will therefore gain energy. So its potential will be greater than zero when it finally arrives in its place in the field.

**Hence:**

**Example 2**

**What's the potential halfway between these?**

**Answer 2**

Consider potential due to A first. Ignore B - we'll deal with it later.

Now consider B and ignore A.

At this mid point, the potential is therefore

It cancels to produce **zero potential**. So moving from infinity to this point would take no energy because the work done by B attracting a positive charge will be equal but opposite to the work done on the charge by A, repelling it.