core39 min studyLesson 20 in Physics

Gauge invariance and electromagnetic stress

Keep observable fields invariant under potential changes and calculate mechanical momentum transfer from electromagnetic stress.

Before you begin

On this page

The question for today

What you’ll understand

Transform scalar and vector potentials together without changing physical fields.
Interpret Maxwell stress as directional momentum transfer.
Calculate radiation pressure while accounting for reflected momentum.

Potential descriptions can change while every electric and magnetic measurement remains the same. Mechanical forces nevertheless remain definite, because fields carry momentum and transfer it to matter. Bringing these ideas together clarifies two different kinds of bookkeeping: gauge freedom removes redundant descriptive choices, while electromagnetic stress accounts for measurable force crossing the boundary of a selected region.

Both potentials must transform together

In time-dependent electromagnetism, the electric field is $-\nabla\phi-\partial_t\mathbf A$ , rather than minus the scalar-potential gradient alone. Adding a gradient to vector potential must therefore be accompanied by subtracting the time derivative of the same gauge function from scalar potential. The two changes cancel in E, while the curl of the added gradient vanishes in B. Forgetting the scalar change creates a physically different electric field.

A gauge condition such as Coulomb or Lorenz gauge selects convenient representatives but need not remove all residual freedom without boundary information. Observable classical force, energy flux, and stress are built from fields and remain unchanged by a legitimate gauge transformation. This redundancy is conceptually different from rotating an apparatus or changing its charge distribution: those operations can change measured fields relative to the apparatus.

\mathbf E=-\nabla\phi-\partial_t\mathbf A,\quad \mathbf B=\nabla\times\mathbf A,\qquad \mathbf A'=\mathbf A+\nabla\chi,\quad \phi'=\phi-\partial_t\chi

χ must have sufficient differentiability for mixed derivatives to commute; its SI units are V·s. These paired transformations leave the electric and magnetic fields unchanged in the domain where the potentials and gauge function are defined.↔ Scroll wide equations horizontally.

The field transports momentum as well as energy

The force density on matter is charge density times electric field plus current density crossed with magnetic field. Maxwell's equations rewrite that force as the divergence of a stress tensor minus the rate of change of field momentum density. The stress tensor describes how a surface with a chosen normal transmits different components of force. It is not a scalar pressure that acts identically in every direction.

For a fixed volume, integrate the stress tensor over its closed boundary and subtract the change in electromagnetic momentum stored inside. The result is the total mechanical force on the charges and currents within. Dropping the stored-momentum term is justified in an appropriate steady or cycle-averaged state, but can fail during a transient. Signs also depend on using an outward surface normal consistently.

T_{ij}=\epsilon_0\!\left(E_iE_j-\frac12\delta_{ij}E^2\right)+\frac1{\mu_0}\!\left(B_iB_j-\frac12\delta_{ij}B^2\right),\quad \mathbf F_{\mathrm{matter}}=\oint T\cdot\hat{\mathbf n}\,dA-\frac d{dt}\int\epsilon_0\mathbf E\times\mathbf B\,dV

This is the vacuum Maxwell stress convention, with T in pascals and ε₀E×B the field momentum per volume. For matter, use an enclosing surface in vacuum or explicitly include the material momentum and constitutive response.↔ Scroll wide equations horizontally.

Reflection doubles normal momentum transfer

A plane wave in vacuum carries momentum equal to its energy divided by c in its propagation direction. A stationary ideal absorber at normal incidence takes the incoming momentum, whereas a perfect reflector reverses the outgoing normal momentum. The latter change is twice as large at the same incident power. This is a flux calculation: a tiny force can persist continuously even though the mirror does not store all arriving radiation energy.

The simple pressure formulas assume normal incidence, a stationary surface, and specified incident intensity. An oblique beam needs projections of both momentum and intercepted area. A heating object can later emit thermal radiation and recoil, so its complete long-term force also depends on the emitted angular distribution. Similarly, a partly transmitting film requires both reflected and transmitted momentum, not an arbitrary interpolation based only on visual brightness.

\mathbf g=\frac{\mathbf S}{c^2},\qquad p_{\mathrm{abs}}=\frac Ic,\qquad p_{\mathrm{refl}}=\frac{2I}{c}\quad\text{(normal incidence in vacuum)}

g is electromagnetic momentum density; p is mechanical pressure in N/m². I is incident cycle-averaged power per area normal to the beam. The absorber formula excludes an additional directed re-emission recoil.↔ Scroll wide equations horizontally.

Work it through

Radiation force on a small mirror

A normally incident vacuum beam of intensity 600 W/m² fully covers a stationary perfect mirror of area 0.020 m². Find its mean pressure and force, neglecting all other forces and taking c = 3.0 × 10⁸ m/s.

The intercepted power is $P=IA=(600)(0.020)=12\,\mathrm W$ . This is the incoming energy per second, not the force and not an amount of energy stored permanently in the mirror.
Reflection reverses each outgoing normal momentum, giving mean pressure $p=2I/c=1200/(3.0\times10^8)=4.0\times10^{-6}\,\mathrm{Pa}$ .
Multiply by area: $F=pA=(4.0\times10^{-6})(0.020)=8.0\times10^{-8}\,\mathrm N$ . The force is along the incident propagation direction, away from the source.

The result

The ideal mirror feels 4.0 μPa of radiation pressure and an 80 nN force. An equally sized ideal absorber would receive half that direct optical force before any re-emission contribution.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Momentum transfer has a direction

Change the angle and magnitude of one two-dimensional vector and inspect its projections. Interpret its components as a normalized momentum-flow direction before considering a chosen surface. This lab does not calculate the Maxwell stress tensor, reflect a ray, or include the area projection required for oblique radiation pressure.

Vectorx / y projectionsTrace marker

x component: 1.393
y component: 0.975
Magnitude: 1.7

Angle35°Magnitude1.7

t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forSurface force requires directional momentum accounting, not just the magnitude of an energy flux.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The gradient of the scalar-potential change contributes +∇∂tχ to E, and the vector-potential change contributes −∂t∇χ. They cancel for a smooth gauge function. The magnetic field is unchanged because curl ∇χ is zero.

Your turn

Try it yourself

Start from zero potentials and choose $\chi=\alpha xt$ , with constant $\alpha=2\,\mathrm{V/m}$ . Find the transformed potentials and verify the fields, including their units.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Differentiate χ once in space and once in time; use both terms in the electric field.

Compare with the solution

The transformed vector potential is $\mathbf A'=(\alpha t,0,0)$ in V·s/m, and $\phi'=-\alpha x$ in volts. Thus $E_x'=-\partial_x(-\alpha x)-\partial_t(\alpha t)=\alpha-\alpha=0$ , and all other electric components vanish. The curl of A′ is zero because it has no spatial dependence, so B′ also vanishes. Nonzero potential values therefore describe the original field-free situation; using only the scalar gradient would incorrectly predict a field.

Keep the idea

Take it with you

Time-dependent gauge transformations must change both scalar and vector potentials.
Maxwell stress and stored field momentum together determine mechanical force.
Radiation pressure follows momentum change and the optical boundary conditions.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.