Classical Electrodynamics · Problem 6.10

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · angular-momentum-conservation-macroscopic-medium

Problem

With the same assumptions as in Problem 6.9 discuss the conservation of angular momentum. Show that the differential and integral forms of the conservation law are

\frac{\partial}{\partial t}(\mathscr{L}_{\text{mech}} + \mathscr{L}_{\text{field}}) + \nabla \cdot \overleftrightarrow{\mathbf{M}} = 0

and

\frac{d}{dt}\int_V (\mathscr{L}_{\text{mech}} + \mathscr{L}_{\text{field}})\, d^3x + \int_S \mathbf{n} \cdot \overleftrightarrow{\mathbf{M}}\, da = 0

where the field angular-momentum density is

\mathscr{L}_{\text{field}} = \mathbf{x} \times \mathbf{g} = \mu\epsilon\, \mathbf{x} \times (\mathbf{E} \times \mathbf{H})

and the flux of angular momentum is described by the tensor

\overleftrightarrow{\mathbf{M}} = \overleftrightarrow{\mathbf{T}} \times \mathbf{x}

Note: Here we have used the dyadic notation for

M_{ij}

and

T_{ij}

. A double-headed arrow conveys a fairly obvious meaning. For example,

\mathbf{n} \cdot \overleftrightarrow{\mathbf{M}}

is a vector whose

j

th component is

\Sigma_i n_i M_{ij}

. The second-rank

\overleftrightarrow{\mathbf{M}}

can be written as a third-rank tensor,

M_{ijk} = T_{ij}x_k - T_{ik}x_j

. But in the indices

j

and

k

it is antisymmetric and so has only three independent elements. Including the index

i

M_{ijk}

therefore has nine components and can be written as a pseudotensor of the second rank, as above.

Related theory

Conservation Of Angular Momentum

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