Question

A conducting disk of a radius a and a small height h is made of a material
of a finite conductivity σ and a permeability o μ . It is placed on the xyplane
in the presence of a uniform, time-varying, magnetic flux density
B = azBo cos ωt as shown in Fig. 6.14. Ignoring time-delay of emf at
different points on the disk, and neglecting the magnetic field induced by
the current in the disk, compute
(a) induced emf along a circle of radius ρ,
(b) total time-average power dissipated in the disk.

354 6 Time-Varying Fields and Maxwell's Equations 6-6 A conducting disk of a radius a and a small height h is made of a material of a finite conductivity and a permeability ,. It is placed on the xy- plane in the presence of a uniform, time-varying, magnetic flux density Bra B. cos ot as shown in Fig. 6.14. Ignoring time-delay of emf at different points on the disk, and neglecting the magnetic field induced by the current in the disk, compute (a) induced emf along a circle of radius p. (b) total time-average power dissipated in the disk. dp Fig. 6.14 A conducting disk in a time varying field (Problem 6-6).

Answer 1

(a) According to Faraday's Law

$\varepsilon =-\frac{d\Phi _{B}}{dt}=-\pi \rho ^{2}\frac{d}{dt}(a_{z}B_{0}\cos \omega t)=\pi \rho ^{2}\omega a_{z}B_{0}\sin \omega t$

(b) The power dissipated in the disk can be calculated using:

$\frac{P}{Vol}=\sigma E^{2}$

The E field can be obtained from the previous part

$\varepsilon =\oint \bar{E}\cdot \bar{dl}=E2\pi \rho$

Then

$E=\frac{\rho\omega a_{z}B_{0}\sin \omega t}{2}$

We can consider the power dissipated in an infinitesimal ring volume and integrate over the disk

$dP=\sigma E^{2}dV=\frac{\sigma \rho^{2}\omega^{2} a^{2}_{z}B^{2}_{0}\sin^{2} \omega t}{4}\times h2\pi \rho d\rho$

$P=\frac{\pi h\sigma \omega^{2} a^{2}_{z}B^{2}_{0}\sin^{2} \omega t}{2}\int_{0}^{a} \rho^{3} d\rho$

$P=\frac{\pi h\sigma \omega^{2} a^{2}_{z}B^{2}_{0}\sin^{2} \omega t a^{4}}{8}$

Finally since the time averaged value of sin² is equal to 1/2 we get

$P_{avg}=\frac{\pi h\sigma \omega^{2} a^{2}_{z}B^{2}_{0} a^{4}}{16}$