Energy and momentum in electromagnetic waves

Electromagnetic waves carry energy through space and exert forces when they transfer momentum to matter. Sunlight warms a surface because field energy is absorbed. Light pressure can move small objects or spacecraft sails because the absorbed or reflected wave changes momentum.

The field description is local. Energy density says how much electromagnetic energy is stored per volume. The Poynting vector says the direction and rate at which energy crosses an area. Momentum density and radiation pressure connect that energy flow to mechanical effects.

Electromagnetic energy density

In vacuum, the electric-field energy density is

\[u_E=\frac{1}{2}\varepsilon_0E^2\]

and the magnetic-field energy density is

\[u_B=\frac{B^2}{2\mu_0}\]

The total is

\[u=u_E+u_B =\frac{1}{2}\left( \varepsilon_0E^2+ \frac{B^2}{\mu_0} \right)\]

Its SI unit is joules per cubic metre. These formulas apply to the instantaneous local fields, not only to sinusoidal waves.

For a vacuum plane wave, \(E=cB\) and \(c^2=1/(\mu_0\varepsilon_0)\). Therefore

\[u_E=u_B\]

and the total plane-wave energy density can be written

\[u=\varepsilon_0E^2=\frac{B^2}{\mu_0}\]

The equality of electric and magnetic parts is a property of a traveling vacuum plane wave, not every electromagnetic field. A static capacitor and a static inductor do not need equal energy densities.

The Poynting vector

Energy flux is described by the Poynting vector

\[\mathbf S=\frac{1}{\mu_0}\mathbf E\times\mathbf B\]

Its units are watts per square metre. The cross product points in the direction of energy transport. For a plane wave with electric field in positive y and magnetic field in positive z, energy travels in positive x.

Power crossing an oriented surface is

\[P=\int_A\mathbf S\cdot d\mathbf A\]

Positive dot product means energy crosses with the chosen surface normal. For a closed surface, outward normals make positive total flux mean energy leaves the enclosed volume.

Poynting's theorem

Local electromagnetic energy conservation is

\[\frac{\partial u}{\partial t} +\boldsymbol\nabla\cdot\mathbf S =-\mathbf J\cdot\mathbf E\]

The term \(\mathbf J\cdot\mathbf E\) is the rate per volume at which the field does work on charges. Integrating over a fixed volume gives

\[\frac{dU_{\mathrm{field}}}{dt} +\oint_{\partial V}\mathbf S\cdot d\mathbf A =-\int_V\mathbf J\cdot\mathbf E\,dV\]

Field energy can decrease because it flows outward, because it does work on matter, or both. Calling every decrease “numerical loss” would be as wrong as ignoring numerical loss entirely.

initial_energy, final_energy = 2.0, 1.5
outward_energy, work_on_matter = 0.3, 0.2
residual = final_energy - initial_energy + outward_energy + work_on_matter
print("Poynting energy balance (J):", residual)
assert abs(residual) < 1e-12

All three terms must cover the same time interval and system boundary.

Instantaneous and average intensity

For a sinusoidal vacuum plane wave,

\[E(t)=E_0\cos(kx-\omega t)\]

the instantaneous Poynting magnitude is

\[S(t)=\varepsilon_0cE_0^2 \cos^2(kx-\omega t)\]

The intensity is the time-averaged energy flux:

\[I=\langle S\rangle =\frac{1}{2}\varepsilon_0cE_0^2 =\frac{E_0B_0}{2\mu_0}\]

Using root-mean-square electric field,

\[I=\varepsilon_0cE_{\mathrm{rms}}^2, \qquad E_{\mathrm{rms}}=\frac{E_0}{\sqrt2}\]

Do not insert a peak amplitude into the RMS formula or apply the factor of one-half twice.

Worked intensity example

For electric-field amplitude \(E_0=300\ \mathrm{V/m}\), the average vacuum intensity is

\[I=\frac{1}{2} (8.854\times10^{-12}\ \mathrm{F/m}) (2.998\times10^8\ \mathrm{m/s}) (300\ \mathrm{V/m})^2 \approx119\ \mathrm{W/m^2}\]

Doubling the field amplitude multiplies intensity by four because energy density and Poynting magnitude depend on squared field amplitude.

Momentum density

An electromagnetic field carries momentum density. In vacuum,

\[\mathbf g=\frac{\mathbf S}{c^2}\]

For a plane wave, energy \(U\) traveling in one direction carries momentum magnitude

\[p=\frac{U}{c}\]

This classical field relation agrees with the quantum photon relation when total energy is the sum of photon energies.

Radiation pressure

When a wave is normally incident on a perfectly absorbing surface, its momentum is transferred to the surface. Average radiation pressure is

\[p_{\mathrm{rad}}=\frac{I}{c}\]

For a perfectly reflecting surface at normal incidence, the momentum reverses, producing twice the change:

\[p_{\mathrm{rad}}=\frac{2I}{c}\]

Here the symbol \(p_{\mathrm{rad}}\) means pressure, not momentum. Its units are pascals. For partial absorption, reflection, oblique incidence, or transmission, momentum components and material response must be accounted for explicitly.

The force on an illuminated area \(A\) under uniform normal pressure is

\[F=p_{\mathrm{rad}}A\]

The force is usually small at ordinary light intensity but becomes important in precision measurements, intense lasers, astrophysics, and solar sails.

Momentum direction and stress

The Poynting direction describes energy flow for many wave situations, but force on a surface can depend on the full electromagnetic stress, particularly for oblique fields or near-field configurations. The Maxwell stress tensor organizes momentum flux and surface traction. Radiation-pressure formulas above are special plane-wave limits, not universal substitutes for that tensor.

Standing waves also show why local interpretation requires care. Opposite traveling waves can produce zero average net Poynting flux while maintaining nonzero, spatially varying field energy. Energy can oscillate locally between electric and magnetic forms even when there is no net long-term transport through a chosen plane.

Discrete field ledger

For sampled fields, compute energy in each cell using its physical volume and compute boundary flux with oriented face areas. Field components may be staggered in space or time; interpolate them consistently before forming products.

c, epsilon, E = 299792458.0, 8.8541878128e-12, 10.0
mu = 1 / (epsilon * c * c)
B, volume = E / c, 0.01
electric_density = 0.5 * epsilon * E**2
magnetic_density = 0.5 * B**2 / mu
energy = (electric_density + magnetic_density) * volume
print("Instantaneous wave energy in volume (J):", energy)
print("Instantaneous flux (W/m^2):", E * B / mu)

A source-free periodic domain should have zero net boundary transfer. An open domain may lose interior energy correctly when outward flux accounts for it. Reduce the grid and time spacing to distinguish discretization drift from physical transfer.

Verification and limits

For a vacuum plane wave, verify equal electric and magnetic energy densities, Poynting direction parallel to propagation, and average intensity proportional to amplitude squared. Confirm that an absorbing surface receives momentum rate equal to incident power divided by light speed, and a perfect reflector receives twice that normal rate.

The vacuum formulas require modification in dispersive, absorbing, anisotropic, nonlinear, or moving media. Material energy can reside in polarization, magnetization, heat, or mechanical motion. Finite grids add interpolation error, numerical dispersion, and boundary artifacts. A small global energy residual does not prove local fields or momentum transfer are accurate.

Local accounting should accompany the global total. The Poynting theorem relates the time rate of electromagnetic energy density to flux divergence and work on charge. A global sum can hide equal and opposite local errors, so compare the discrete balance in individual cells as well as over the whole domain, especially near sources and boundaries.

A worked Python notebook

Relate the peak electric amplitude of a sinusoidal plane wave to its time-averaged intensity. The question for this calculation is precise: how does mean vacuum intensity change when we change peak electric field, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The factor of one half comes from averaging the squared sinusoid over a cycle. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 100 V/m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for peak electric field, expressed in V/m. The output is mean vacuum intensity, expressed in W/m^2. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 0.00132720936399648 q^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 10 V/m, the expression gives 0.132721 W/m^2. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("8.8541878128e-12*299792458*q**2/2", locals={"q": q})
reference_input = 10.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "V/m")
print("Predicted mean vacuum intensity:", reference_output, "W/m^2")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 W/m^2. The midpoint, at 50 V/m, gives 3.31802 W/m^2. The upper endpoint gives 13.2721 W/m^2. Read these together with the assumptions: they describe different choices of peak electric field within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 50.0, 100.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} V/m -> {output_value:.6g} W/m^2")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows peak electric field; the vertical axis shows mean vacuum intensity. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 100.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Peak electric field (V/m)")
ax.set_ylabel("Mean vacuum intensity (W/m^2)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does mean vacuum intensity change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10 V/m is 0.0265442 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 0.00265441872799296 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 10.5
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "W/m^2")
print("Full expression:", full_prediction, "W/m^2")
print("Approximation difference:", full_prediction - linear_prediction, "W/m^2")

Here the changed input is 10.5 V/m. The tangent estimate is 0.145993 W/m^2, while the full expression gives 0.146325 W/m^2. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 100.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of peak electric field inside the stated interval, predict whether mean vacuum intensity will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The factor of one half comes from averaging the squared sinusoid over a cycle. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of W/m^2: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(100.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 100.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 2, Section 16.3, “Energy Carried by Electromagnetic Waves”.
• OpenStax, University Physics Volume 2, Section 16.4, “Momentum and Radiation Pressure”.
• Benjamin Crowell, Light and Matter.
• Ralph A. Llewellyn and Paul A. Tipler, Modern Physics.
• Physics LibreTexts, “Electricity and Magnetism”.