Simple Nature - Light and Matter

d/Positive and negative signs in

Ampere’s law.`

e/Example 13: a cutaway

view of a solenoid.

the products of the vectors’ magnitudes. The resulting circulation

is

Γ =|s 1 ||B 1 |+|s 2 ||B 2 |

=

2 πkηs

c^2

+

2 πkηs

c^2

=

4 πkηs

c^2

.

Butηsis (current/length)(length), i.e., it is the amount of current
that pierces the Amperian surface. We’ll call this currentIthrough. We have found one specific example of the general law of nature known as Ampere’s law:

Γ =

4 πk

c^2

Ithrough

Positive and negative signs
Figures d/1 and d/2 show what happens to the circulation when
we reverse the direction of the currentIthrough. Reversing the cur-
rent causes the magnetic field to reverse itself as well. The dot
products occurring in the circulation are all negative in d/2, so the
total circulation is now negative. To preserve Amp`ere’s law, we need
to define the current in d/2 as a negative number. In general, deter-
mine these plus and minus signs using the right-hand rule shown in
the figure. As the fingers of your hand sweep around in the direction
of thesvectors, your thumb defines the direction of current which is
positive. Choosing the direction of the thumb is like choosing which
way to insert an ammeter in a circuit: on a digital meter, reversing
the connections gives readings which are opposite in sign.

A solenoid example 13

.What is the field inside a long, straight solenoid of length`and

radiusa, and havingNloops of wire evenly wound along it, which

carry a currentI?

.This is an interesting example, because it allows us to get a

very good approximation to the field, but without some experi-

mental input it wouldn’t be obvious what approximation to use.

Figure e/1 shows what we’d observe by measuring the field of a

real solenoid. The field is nearly constant inside the tube, as long

as we stay far away from the mouths. The field outside is much

weaker. For the sake of an approximate calculation, we can ideal-

ize this field as shown in figure e/2. Of the edges of the Amperian`

surface shown in e/3, only AB contributes to the flux — there is

zero field along CD, and the field is perpendicular to edges BC

Section 11.3 Magnetic fields by Ampere’s law` 701