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Maxwell's equations symbolically

Maxwell's equations connect electric charge, electric current, electric field, and magnetic field. Together with the Lorentz force law, they form the foundation of classical electromagnetism. Their compact symbols encode electrostatics, magnetostatics, induction, electromagnetic waves, and local charge conservation.

The equations can be written in integral or differential form. Integral equations relate flux or circulation over finite surfaces and paths. Differential equations describe the same laws locally at each smooth point. Boundary and singular sources require particular care when moving between the two forms.

Symbols and SI units

The principal fields and sources are:

The equations below use SI units in vacuum. Matter can be described using polarization, magnetization, and constitutive relations, or with auxiliary fields under a clearly stated convention.

Gauss's law for electricity

The integral form is

\[\oint_S\mathbf E\cdot d\mathbf A =\frac{Q_{\mathrm{enclosed}}}{\varepsilon_0}\]

Electric flux through any closed surface equals enclosed charge divided by vacuum permittivity. Charge outside the surface can contribute to the field at individual points, but its net flux through the closed surface is zero.

The differential form is

\[\boldsymbol\nabla\cdot\mathbf E =\frac{\rho}{\varepsilon_0}\]

It says charge density is a local source of electric-field divergence. For a point charge, charge density contains a delta distribution, so the ordinary smooth equation must not be evaluated naively at the singular point.

Gauss's law for magnetism

The magnetic equation is

\[\oint_S\mathbf B\cdot d\mathbf A=0\]

or locally,

\[\boldsymbol\nabla\cdot\mathbf B=0\]

There are no observed isolated magnetic charges in classical electromagnetism. Magnetic field lines do not begin or end; they form closed loops or extend through the boundary of the modeled region. A grid calculation with nonzero magnetic divergence has introduced an unphysical numerical source unless the model explicitly extends Maxwell's equations.

Faraday's law

Changing magnetic flux produces a circulating electric field:

\[\oint_C\mathbf E\cdot d\boldsymbol\ell =-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf A\]

The path direction and surface normal follow the right-hand convention. The negative sign encodes Lenz's law. Locally,

\[\boldsymbol\nabla\times\mathbf E =-\frac{\partial\mathbf B}{\partial t}\]

An electrostatic field has zero curl, but an induced electric field generally does not. Therefore a single-valued electrostatic potential cannot globally describe a region threaded by changing magnetic flux.

Ampère–Maxwell law

The complete circulation law in vacuum is

\[\oint_C\mathbf B\cdot d\boldsymbol\ell =\mu_0I_{\mathrm{enclosed}} +\mu_0\varepsilon_0 \frac{d\Phi_E}{dt}\]

The second term is the displacement-current contribution from changing electric flux. Locally,

\[\boldsymbol\nabla\times\mathbf B =\mu_0\mathbf J +\mu_0\varepsilon_0 \frac{\partial\mathbf E}{\partial t}\]

For steady fields, the time-derivative term vanishes and the equation reduces to the magnetostatic Ampère law. For changing fields, omitting displacement current breaks charge conservation and makes the answer depend on which surface spans a chosen loop.

Why displacement current is necessary

Consider a charging capacitor. Conduction current flows in the wires, but no ordinary charge carriers cross the insulating gap. A surface cutting the wire encloses conduction current, while a surface stretched through the gap does not. Both surfaces share the same boundary loop, so magnetic circulation cannot depend on which surface is imagined.

As charge accumulates on the plates, electric flux in the gap changes. For an ideal parallel-plate capacitor, its displacement-current term equals the conduction current:

\[I_{\mathrm d} =\varepsilon_0\frac{d\Phi_E}{dt}=I\]

Maxwell's addition makes the circulation equation consistent across both surfaces.

Charge conservation follows from Maxwell's equations

Take the divergence of the differential Ampère–Maxwell equation. The divergence of any curl is zero, so

\[0=\mu_0\boldsymbol\nabla\cdot\mathbf J +\mu_0\varepsilon_0 \frac{\partial}{\partial t} \left(\boldsymbol\nabla\cdot\mathbf E\right)\]

Substitute Gauss's electric law to obtain

\[\frac{\partial\rho}{\partial t} +\boldsymbol\nabla\cdot\mathbf J=0\]

This identity is not an optional extra. Charge and current sources supplied to a field model must satisfy it. Otherwise no exact electromagnetic solution can satisfy all four Maxwell equations simultaneously.

From integral to differential form

The divergence theorem states

\[\oint_S\mathbf F\cdot d\mathbf A =\int_V\boldsymbol\nabla\cdot\mathbf F\,dV\]

and Stokes' theorem states

\[\oint_C\mathbf F\cdot d\boldsymbol\ell =\int_S\left(\boldsymbol\nabla\times\mathbf F\right) \cdot d\mathbf A\]

Applying these theorems to arbitrary small volumes or surfaces connects the integral and differential Maxwell equations. The argument assumes sufficient smoothness away from singular sources and material boundaries. Across a boundary, the integral forms are often safer because they directly produce jump conditions for field components.

Source-free wave equations

In vacuum without charge or current, \(\rho=0\) and \(\mathbf J=0\). Take the curl of Faraday's law and substitute the Ampère–Maxwell equation. Using the vector identity

\[\boldsymbol\nabla\times \left(\boldsymbol\nabla\times\mathbf E\right) =\boldsymbol\nabla \left(\boldsymbol\nabla\cdot\mathbf E\right) -\nabla^2\mathbf E\]

and the source-free divergence constraint gives

\[\nabla^2\mathbf E -\mu_0\varepsilon_0 \frac{\partial^2\mathbf E}{\partial t^2}=0\]

The magnetic field satisfies the same form. The wave speed is

\[c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}\]

This agreement with the measured speed of light identifies light as an electromagnetic wave.

Potentials and gauge freedom

Because magnetic divergence is zero, the magnetic field can be written as the curl of a vector potential:

\[\mathbf B=\boldsymbol\nabla\times\mathbf A\]

Faraday's law is then satisfied by

\[\mathbf E=-\boldsymbol\nabla V -\frac{\partial\mathbf A}{\partial t}\]

The potentials are not unique. Different gauge choices can produce the same electric and magnetic fields. A numerical method using potentials must specify its gauge constraint; otherwise it can contain redundant degrees of freedom or a singular system.

Symbolic diagnostic residuals

For discrete fields, define a residual for each equation rather than assuming that a finite-difference operator preserves the continuum identities exactly.

import sympy as sp

x, t, k, c = sp.symbols("x t k c", positive=True)
E = sp.cos(k * x - c * k * t)
B = E / c
# E points along y, B along z; propagation is along +x.
faraday_residual = sp.diff(E, x) + sp.diff(B, t)
ampere_residual = -sp.diff(B, x) - sp.diff(E, t) / c**2
assert sp.simplify(faraday_residual) == 0
assert sp.simplify(ampere_residual) == 0
print("Vacuum curl-equation residuals:", faraday_residual, ampere_residual)
# Divergences vanish because Ey and Bz depend only on x and t.

Each residual has units. Report a scale-aware norm, maximum magnitude, location of that maximum, grid spacing, and boundary policy. A low residual for one equation does not imply the other three pass.

Verification and limits

Uniform source-free fields should have zero discrete divergence and curl away from boundaries. A static point-charge model should satisfy flux proportional to enclosed charge, with the singular source represented consistently. A plane wave should be transverse, propagate at the expected speed, and satisfy both curl equations. Source data must pass the continuity equation.

Vacuum Maxwell equations do not by themselves describe material polarization, magnetization, conductivity, dispersion, nonlinearity, or microscopic quantum behavior. Finite grids add numerical dispersion, boundary reflection, source-deposition error, and roundoff. The symbolic equations define what must be checked; they do not guarantee that one discretization preserves every identity automatically.

A worked Python notebook

At time zero, inspect one component of a vacuum plane wave with wavelength one in scaled units. The question for this calculation is precise: how does transverse electric field change when we change position, 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 companion magnetic field and time dependence must satisfy both curl equations. 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 1 scaled length units. 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 position, expressed in scaled length units. The output is transverse electric field, expressed in scaled field units. 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.

\[f(q) = \cos{\left(2 \pi q \right)}\]

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 0.25 scaled length units, the expression gives 0 scaled field units. 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("cos(2*pi*q)", locals={"q": q})
reference_input = 0.25
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "scaled length units")
print("Predicted transverse electric field:", reference_output, "scaled field units")

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 1 scaled field units. The midpoint, at 0.5 scaled length units, gives -1 scaled field units. The upper endpoint gives 1 scaled field units. Read these together with the assumptions: they describe different choices of position 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, 0.5, 1.0])
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} scaled length units -> "
        f"{output_value:.6g} scaled field units"
    )
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 position; the vertical axis shows transverse electric field. 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, 1.0, 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("Position (scaled length units)")
ax.set_ylabel("Transverse electric field (scaled field units)")
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 transverse electric field change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.25 scaled length units is -6.28319 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - 2 \pi \sin{\left(2 \pi q \right)}\]

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 = 0.255
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, "scaled field units")
print("Full expression:", full_prediction, "scaled field units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "scaled field units",
)

Here the changed input is 0.255 scaled length units. The tangent estimate is -0.0314159 scaled field units, while the full expression gives -0.0314108 scaled field units. 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 = 1.0
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 position inside the stated interval, predict whether transverse electric field 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 companion magnetic field and time dependence must satisfy both curl equations. 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 scaled field units: 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(1.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 1.0, 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

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