A uniform, stationary conductor of constant permeability and conductivity obeys the…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Partial differential equations and boundary-value problems

The Boundary-Value Problem, Ch The quasi-static approximation, eddy currents and the skin effect

Problem

A uniform, stationary conductor of constant permeability

\mu

and conductivity

\sigma

obeys the quasi-static equations

\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t

\nabla\times\mathbf{B} = \mu\sigma\mathbf{E}

and

\nabla\cdot\mathbf{B} = 0

(the displacement current has been dropped). ((a)) Starting from these three equations, derive the magnetic diffusion equation

\nabla^2\mathbf{B} = \mu\sigma\,\partial\mathbf{B}/\partial t

. ((b)) The conductor occupies the half-space

z > 0

and its surface at

z=0

is driven so that

\mathbf{B}(0,t) = B_0\cos(\omega t)\hat{\mathbf{y}}

. Writing

\mathbf{B} = \hat{\mathbf{y}}\,\mathrm{Re}[\tilde{B}(z)e^{-i\omega t}]

and trying

\tilde{B}(z) = B_0e^{i\kappa z}

, find the equation obeyed by

\kappa

, solve it using

\sqrt{i} = (1+i)/\sqrt{2}

, discard the root that is unphysical for

z>0

, and show that

\mathbf{B}(z,t) = B_0e^{-z/\delta}\cos(z/\delta - \omega t)\hat{\mathbf{y}}

with

\delta = \sqrt{2/\mu\sigma\omega}

. ((c)) Now consider a sheet of thickness

d

with

d \ll \delta

, lying in the

yz

-plane, carrying the same field

B_0\cos\omega t\,\hat{\mathbf{y}}

but now taken as uniform across the thickness because

d\ll\delta

. Using Faraday's law on a rectangular loop in the

xz

-plane that reaches

\pm x

from the mid-plane, find

E_z(x)

, form the Ohmic dissipation per unit volume

\sigma E_z^2

, average over one cycle and over the thickness, and show that the eddy-current loss per unit volume is

\sigma\omega^2B_0^2d^2/24

Browse the problem library