Application of Ampere's Law - Magnetic Field of a Solenoid

Magnetic Field of a Solenoid:

We can readily find examples where Ampere's Law is more useful, by virtue of being easier to apply, than the Biot-Savart Law. For example, as was the case for the electric field, it was deemed to be very practical to have a device which could "store" magnetic field as a capacitor "stores" an electric field. We know that what the capacitor really stores is electric charge separation, but one of the properties for which a parallel-plate capacitor is useful is its ability to produce a uniform electric field between its plates. The equivalent device for magnetic fields is the solenoid. It's simply a conducting wire wrapped into a cylindrical shape. Even though the actual shape of the wire is helical, for densely packed wrapping we can actually consider the solenoid to be a bunch of closely spaced coils. Near the center of each coil we know that the magnetic field is nearly perpendicular to the plane of the coil.

The magnetic field near the center of a single coil carrying a current I.

The magnetic field near the center of a set of three coils all carrying a current I.

If we place coils on either side of the first and let them all carry the same current I in the same direction, then the magnetic field lines will link together near the center to make a field that is approximately uniform in direction and strength near the center of the set of coils. Placing many coils in close proximity to each other yields the solenoid field.

A solenoid approximates many current-carrying coils spaced much more closely than the coil diameter.

A perfect solenoid has coils so close together that the magnetic field is zero outside the solenoid and perfectly uniform inside.

Approximating the field as constant in direction and magnitude near the center of the solenoid allows us to use Ampere's Law to calculate its magnitude.

Applying Ampere's Law to a solenoid.

Choosing a flat rectangle as the Amperian loop, we see that the contributions to the loop integral can be broken into four parts. The area bounded by the flat rectangle is penetrated by N loops over a length L.

where n = N/L is the linear density of current loops.