Simple Nature - Light and Matter

Potential associated with a point charge example 10

.What is the potential associated with a point charge?

.As derived previously in self-check A on page 583, the field is

|E|=

k Q

r^2

The difference in potential between two points on the same radius

line is

∆V=−

∫

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 casexreally meansr.

∆V=−

∫r 2

r 1

Erdr

=−

∫r 2

r 1

k Q

r^2

dr

=

k Q

r

]r 2

r 1

=

k Q

r 2

−

k Q

r 1

.

The standard convention is to user 1 =∞as a reference point, so

that the potential at any distancerfrom the charge is

V=

k Q

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 B

Show that you can recover the expression for the field of a point charge

by evaluating the derivativeEx=−dV/dx. .Answer, p. 1058

Weighing an electron example 11

J.J. Thomson (p. 489) is considered to have discovered the elec-

tron because he measured its charge-to-mass ratioq/mand found

it to be much larger than that of an ionized atom, interpreting this

as evidence that he was seeing a subatomic particle with a mass

much small than an atom’s. But not only is the electron’sq/m

relatively large compared to that of an atom, it is simply a huge

number (∼− 1011 C/kg) when expressed in SI units. SI units are

designed for human scales of experience, so this suggests that in

592 Chapter 10 Fields