[NOTES/EM-07013] Conservtion of Electromagnetic Field Momentum.*

There is no equation of continuity for momentum conservation in particle mechanics. This is expected because the laws of Newtonian mechanics are not covariant under Lorentz transformations.

The Maxwell's equations are covariant under Lorentz transformations. The momentum conservation law takes the form of equation of continuity. A conservation law in relativistic  theory involves a current four vector, (volume integral of) the time component gives conserved quantity. The physical interpretation of  the space components is derived using the local nature of conservation law.

In case of momentum conservation for the electromagnetic field, the space part of  the current is stress tensor. The surface integral of the stress tensor equals the flow of momentum across a surface per second and is therefore, equal to the, force on the surface.

We briefly discuss, stating only the results here, the interpretation of the 'current' associated with momentum conservation.
Talking about momentum conservation, the electromagnetic field carries momentum \(\vec {\mathcal P} \) which is conserved for free fields.
\begin{equation}
 \vec {\mathcal P} = \epsilon_0\mu_0 \vec S,
\end{equation}
where \(\vec S= (1/\mu_0) (\vec E \times \vec B)\) is the Poynting vector.
The conservation law for the momentum  takes the form
\begin{equation}
 \pp[{\mathcal P}_j]{t} + \nabla_k\cdot {\mathcal T}_{jk}=0, \quad k=1,2,3.
\end{equation}
Here \({\mathcal T}_{jk}\), are space components of the energy momentum tensor of the electromagnetic field, and are given by
\begin{equation}
 {\mathcal T}_{jk}= \epsilon_0\Big(E_jE_k-\frac{1}{2}\delta_{jk} E^2\Big) + \frac{1}{\mu_0}\Big(B_jB_k -\frac{1}{2}B_jB_k\Big)
\end{equation}

Recalling that the current \(\vec J\) in the continuity equation, gives the flow of charge per unit area across a surface. The rate of flow of charge  across a surface element \(\Delta S\), normal \(\hat n\), is given by \(\vec J \cdot \hat n \Delta S\). \\
Applying the same interpretation to the continuity equation for the e.m. field momentum, we see that \(n_k  {\mathcal T}_{jk}\Delta S\) gives the transfer of momentum across the surface element \(\Delta S\) and hence the \(j^\text{th}\) component of the  force on the surface element.  If we integrate this quantity over a surface \(S\), we would get the force exerted by the electromagnetic waves on the surface \(S\). The tensor \({\mathcal T}_jk\) is called stress tensor as it gives force per unit area, or the stress, acting on a surface in the \(j^\text{th}\) direction.

Reference  Sec 8.2.3  Conservation of Momentum
 David Griffiths, {\it Introduction to Electrodynamics}, 3rd EEE edn, Prentice Hall of India Pvt Ltd New Delhi,  (2002).
    

\(\mathcal P\)