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LM 22.5 Voltage for nonuniform fields Collection

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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

ΔPE=-Fdx,                 [one dimension]

so for electric fields we divide by q to find

ΔV=-Edx,                  [one dimension]

Applying the fundamental theorem of calculus yields

E=-dVdx                   [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|=kQr2

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

              ΔV=dV
              =-Exdx

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.

             ΔV=r2r1Erdx

                     =r2r1kQr2dr

                     =kQr]r2r1

                     ΔV=kQr2-kQr1.

The standard convention is to use r1 = ∞ as a reference point, so that the voltage at any distance r from the charge is

           V=kQr

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 Ex=-dVdx.

(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

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