1. Ampere’s Law

2. Magnetic Field Loop A closed loop can be drawn in a magnetic field.
Any shape path
Include direction along path
Ampere deduced a relationship between the loop and currents. B I

3. Current Through Loop
The loop can be broken down into separate segments Dl.
The field parallel to an individual segment is B║.
Opposite to loop direction is negative
The net current is the sum of all currents into the loop.
Out of loop is negative

4. Ampere’s Law
The sum of the parallel fields is based on the field direction on the perimeter of the loop.
The sum of the currents is based on the area within the loop.
Ampere’s law states that these sums are equal.

5. Application to Line
Apply Ampere’s law to a straight line current in a wire.
Use a loop that follows a circular field line.
B always parallel to Dl
One current through the loop I.
Ampere’s law gives the expected result. B I

6. Solenoid Current
For a field inside the solenoid use four straight lines to make the loop.
The loop length L along the axis defines the number of turns of wire in the loop. NI B L l

7. Solenoid Field
For a field inside the solenoid use four straight lines to make the loop.
B is perpendicular to radial segments.
No contribution to sum
Outer segment has no field if solenoid is long enough.
The only contribution is the inside segment. NI B l

8. Solenoid Match
Ampere’s law can be applied to compare the current and the field.
Independent of choice of line inside the coil NI B l