An induction furnace uses electromagnetic induction to produce eddy currents in a conductor, thereby raising the conductor’s temperature. Commercial units operate at frequencies ranging from 60 Hz to about 1 MHz and deliver powers from a few watts to several megawatts. Induction heating can be used for welding in a vacuum enclosure, to avoid oxidation and contamination of the metal. At high frequencies, induced currents occur only near the surface of the conductor—this is the “skin effect.” By creating an induced current for a short time at an appropriately high frequency, one can heat a sample down to a controlled depth. For example, the surface of a farm tiller can be tempered to make it hard and brittle for effective cutting while keeping the interior metal soft and ductile to resist breakage. To explore induction heating, consider a flat conducting disk of radius R, thickness b, and resistivity 3. A sinusoidal magnetic field Bmax cos *t is applied perpendicular to the disk. Assume that the frequency is so low that the skin effect is not important. Assume the eddy currents occur in circles concentric with the disk.
(a) Calculate the average power delivered to the disk.
(b) What If? By what factor does the power change when the amplitude of the field doubles?
(c) When the frequency doubles?
(d) When the radius of the disk doubles?