Part 13: Electromagnetic FIelds and Angular Momentum
Last week, we considered momentum conservation in electromagnetism, finding that the momentum density is equal to 
, where 
is the Poynting vector; and that the momentum flux density tensor is 
, where 
is the Maxwell stress tensor, a symmetric rank-two tensor with components +\frac1{\mu_0}\left(B_iB_j-\frac12B^2\delta_{ij}\right))
. Today, we consider angular momentum.
Recalling that for a particle of momentum p at displacement x from the origin, the angular momentum about the origin is
, and the force on a particle of charge q is 
, so the torque about the origin on the particle is
)
.
Converting to charge and current densities, if 
is the total (mechanical) angular momentum of the charges in a volume V, then
\,dV)
.
However, we showed last week that
)
,
where 
is the divergence of the Maxwell stress tensor , which is a vector with ith component 
.
Thus, we plug this in to get:
\,dV-\int_{V}\mathbf{x}\times\left(\frac{\partial}{\partial{t}}\left(\frac1{c^2}\mathbf{S}\right)\right)\,dV)
.
Note that since 
, \right)=\frac{\partial}{\partial{t}}\left(\mathbf{x}\times\left(\frac1{c^2}\mathbf{S}\right)\right))
,
and the order can be interchanged between time derivatives and volume integrals, so the above becomes:
\,dV=\int_{V}\mathbf{x}\times\left(\mathbf{\nabla}\cdot\mathbb{T}\right)\,dV)
,
and so we see \,dV=\mathbf{L}_{em})
is the angular momentum of the fields in the volume, and so =\frac1{c^2}\left(\mathbf{x}\times\mathbf{S}\right))
is the angular momentum density of the electromagnetic fields.
Now, let us consider the flux term. The ith component of the cross product of vectors a and b is _{i}=\sum_{j,k}\epsilon_{ijk}a_jb_k)
, where 
is the Levi-Civita symbol. Thus, the ith component of )
is
\right)_i=\sum_{j,k}\epsilon_{ijk}x_j\left(\mathbf{\nabla}\cdot\mathbb{T}\right)_k=\sum_{j,k,m}\epsilon_{ijk}x_j\frac{\partial}{\partial{x_m}}T_{km})
.
Now, using the product rule, =x_j\frac{\partial}{\partial{x_m}}T_{km}+\left(\frac{\partial}{\partial{x_m}}x_j\right)T_{km})
, and 
, so
-\delta_{jm}T_{km}\right]\\&=&\left[\sum_k\frac{\partial}{\partial{x_m}}\left(x_jT_{km}\right)\right]-T_{kj}\end{eqnarray})
,
and so
\right)_i&=&\sum_{j,k,m}\epsilon_{ijk}\frac{\partial}{\partial{x_m}}\left(x_jT_{km}\right)-\sum_{j,k}\epsilon_{ijk}T_{kj}\\&=&\sum_{m}\frac{\partial}{\partial{x_m}}\left[\sum_{j,k}\epsilon_{ijk}x_jT_{km}\right]\end{eqnarray})
,
since 
due to the symmetry of and the antisymmetry of the Levi-Civita symbol (exchange the labels on the dummy indices j and k, then reverse orders on , with no sign change, and on 
, with sign change, to find that the sum is its own opposite, and therefore zero).
Note that the cross product of vectors a and b is the vector with ith component . Similarly, the cross product of the vector a with the tensor 
is the tensor with components
_{im}=\sum_{j,k}\epsilon_{ijk}a_jB_{km})
.
In this vein, we see then that _{im})
, and therefore, that
\right)_i&=&\sum_{m}\frac{\partial}{\partial{x_m}}\left[\sum_{j,k}\epsilon_{ijk}x_jT_{km}\right]\\&=&\sum_{m}\frac{\partial}{\partial{x_m}}\left(\mathbf{x}\times\mathbb{T}\right)_{im}\\&=&\left[\mathbf{\nabla}\cdot\left(\mathbf{x}\times\mathbb{T}\right)\right]_i\end{eqnarray})
,
the ith component of the divergence of the tensor (or, more accurately, pseudotensor) 
. Thus, since the corresponding components are equal,
=\mathbf{\nabla}\cdot\left(\mathbf{x}\times\mathbb{T}\right)=\mathbf{\nabla}\cdot\mathbb{M})
,
and so
,
giving us our integral continuity equation. Just as is the linear momentum flux density tensor of electromagnetic fields, so we see that 
is the angular momentum flux density tensor of electromagnetic fields. Thus, the differential form for our continuity equation is
=\mathbf{\nabla}\cdot\mathbb{M})
.
And just as one may find the total force the fields exert on the charges in a volume by integrating the total force via 
, one can do the same with 
to find the total torque.
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Tags: Angular Momentum, Continuity Equation, Electricity & Magnetism, Friday Physics, Maxwell Stress Tensor, physics, Tensor, Torque
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