Past exam of the electromagnetism course of the University of Oxford / 2023 / 2 / a / ii

Table of contents
- Solution ii

Solution

Allow time dependence while keeping the constitutive relations linear, isotropic, nondispersive, and independent of time. The Maxwell equations in matter give

\nabla \times E = - \partial_{t} B, \nabla \times H = J_{f} + \partial_{t} D .

For the Poynting vector

P = E \times H

, the divergence identity yields

\nabla \cdot P = H \cdot (\nabla \times E) - E \cdot (\nabla \times H) = - \frac{1}{μ} B \cdot \partial_{t} B - ϵ E \cdot \partial_{t} E - E \cdot J_{f} .

The first two terms are exact time derivatives. Taking the zero-field energy density as zero gives the electromagnetic energy density

E = \frac{1}{2} ϵ ∣ E ∣^{2} + \frac{1}{2 μ} ∣ B ∣^{2} = \frac{1}{2} (E \cdot D + B \cdot H) .

Consequently the Poynting theorem is the continuity equation

\partial_{t} E + \nabla \cdot P = - J_{f} \cdot E .

In integral form,

\frac{d}{d t} \int_{W} E d^{3} r + \oint_{\partial W} P \cdot n d S = - \int_{W} J_{f} \cdot E d^{3} r .

The left side is the rate of increase of stored electromagnetic energy plus outward Poynting flux. The right side accounts for work delivered to the free electric current; in an Ohmic medium

J_{f} \cdot E

is the local rate of Joule heating.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version