Magnetism - Ampere's Law

where ds is an element of length around an arbitrary closed loop "C", called an Amperian loop and the summation is over all currents passing through the loop.

Currents passing "out of" the loop are defined as positive, currents passing into the loop are negative, whereas current which do not pass through the loop are not included in the summation.

Ampere's Law forms part of the second of Maxwell's equations. We will shortly adjust it slightly (following Maxwell), to complete the second of Maxwell's equations. Remember, Gauss's Law was the first of Maxwell's equations.

As a reminder, Gauss's Law appears below,

Note that Gauss's Law involves a surface integral of E over a closed Gaussian surface "S". Ampere's Law involves a line integral around a closed Amperian loop "C".

Gauss's Law is valid for any arbitrary closed Gaussian surface. Similarly Ampere's Law is valid for any closed (Amperian) loop.

Although Ampere's Law is true for any closed loop "C", it is only useful to calculate B for some very symmetric cases, where we already know (from symmetry) some of the properties of B.

Simple Applications

B due to an infinite straight current carrying wire.
- Symmetry argument:
  Since the wire is infinite, we know from the Biot-Savart Law that B is perpendicular to dl and r and thus lines of B must form concentric circles around the current. Also, B can, at most, depend only on the distance from the wire, r.
- Choice of Amperian Loop:
  The Amperian loop is chosen so that B is constant on the loop and in the same direction as ds - that is a circle whose plane is perpendicular to the wire and centered on the wire. This allows us to take B "out of the integral".
- Evaluation of B:
  With the Amperian loop above we have
  
  so that
  
  directed "circumferentially" around the loop, with the sense given by the right-hand-rule described under the Biot-Savart law.

B due to an infinite solenoid
- Symmetry argument:
  Since the solenoid is infinite, we conclude that B is directed along the axis of the solenoid. Also, B can, at most, depend only on the distance from the axis of the solenoid. For an infinite solenoid B = 0 outside the solenoid.
- Choice of Amperian Loop:
  The rectangular Amperian loop (at right) is chosen so that B is constant on the two sides parallel to the solenoid axis and perpendicular to the ds on the other two sides. This allows us to take B "out of the integral".
- Evaluation of B:
  With the Amperian loop above we have
  
  so that
  
  where n is the number of turns per unit length and the field is directed along the solenoid axis as shown. (Use right-hand-rule)
Note that, the above analysis is true only for an infinite solenoid. For a real solenoid it is a good approximation inside, away from the ends. The B field outside is not zero, but much smaller than the field inside.

The shape of the B field due to a solenoid is the same as that of a bar magnet and a magnetic dipole.

What does a clock do when it's hungry ? It goes back four seconds.

Dr. C. L. Davis
Physics Department
University of Louisville
email: c.l.davis@louisville.edu