Magnetic Fields Within a Solenoid
Recall that an ideal solenoid is a tightly turned coil of wire which produces a uniform magnetic field down its center. We will now use Ampere's Law to calculate the magnitude of this central magnetic field.
Beginning on the left side we will use the Amperian path ABCDA.
where B represents the strength of the central magnetic field and represents the length of the line CD.
Completing our solution, we now have
where the ratio represents the total number of loops in the solenoid divided by the length of the solenoid. This value is a constant represented by the variable **n**.
Magnetic Fields Within a Toroid Now let's wrap a solenoid into a circle to form a toroid and examine its magnetic field.
Once again, we will use Ampere's Law to calculate the magnetic field. To begin we will place an Amperian loop along the center of the toroid's coils so that its radius, **r**, lies between the inner and outer radii of the torus, a < r < b.
For r < a and r > b, B = 0. In the case of r < a, no currents are enclosed within the Amperian loop. For r > b, the currents flowing "in" (-z) and the currents flowing "out" (+z) cancel, giving a net enclosed current of zero. |