22.5 Voltage for nonuniform fields by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

## 22.5 ∫ Voltage for nonuniform fields

The calculus-savvy reader will have no difficulty generalizing the field-voltage relationship to the case of a varying field. The potential energy associated with a varying force is

`DeltaPE=-∫Fdx`, [one dimension]

so for electric fields we divide by `q` to find

`DeltaV=-∫Edx`, [one dimension]

Applying the fundamental theorem of calculus yields

`E=-(dV)/(dx)` [one dimension]

##### Example 12: Voltage associated with a point charge

⇒ What is the voltage associated with a point charge?

⇒ As derived previously in self-check A on page 625, the field is

`|E|=(kQ)/r^2`

The difference in voltage between two points on the same radius line is

`DeltaV=∫dV`

`=-∫E_xdx`

In the general discussion above, x was just a generic name for distance traveled along the line from one point to the other, so in this case x really means r.

`DeltaV=int_(r_1)^(r_2)E_rdx`

`=int_(r_1)^(r_2)(kQ)/r^2dr`

`=(kQ)/r]_(r_1)^(r_2)`

`DeltaV=(kQ)/r_2-(kQ)/r_1`.

The standard convention is to use r_{1} = ∞ as a reference point, so that the voltage at any distance r from the charge is

`V=(kQ)/r`

The interpretation is that if you bring a positive test charge closer to a positive charge, its electrical energy is increased; if it was released, it would spring away, releasing this as kinetic energy.

*self-check:*

Show that you can recover the expression for the field of a point charge by evaluating the derivative `E_x =-(dV)/(dx)`.

(answer in the back of the PDF version of the book)

22.5 Voltage for nonuniform fields by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.