Magnetic fields from moving charge

Electric charge produces an electric field whether it is stationary or moving. Charge in motion also produces a magnetic field. In matter, the motion of enormous numbers of charges is summarized as electric current or current density. The magnetic field keeps directional information, so vector orientation is essential from the beginning.

Magnetism is not a separate kind of charge. In special relativity, electric and magnetic fields are parts of one electromagnetic field, and observers in relative motion can divide that field into electric and magnetic parts differently. This lesson starts with the low-speed formulas used in introductory electromagnetism and states where they stop being reliable.

Current is organized charge motion

Electric current is the signed rate at which charge crosses an oriented surface:

\[I=\frac{dQ}{dt}\]

For one kind of carrier with number density \(n\), charge \(q\), and drift velocity \(\mathbf v_d\), the current density is

\[\mathbf J=nq\mathbf v_d\]

More generally, if \(\rho_q\) is a signed volume charge density moving with velocity \(\mathbf v\), then

\[\mathbf J=\rho_q\mathbf v\]

Electrons in a metal drift opposite the conventional current because their charge is negative. The product of their negative charge and opposite drift direction makes the conventional current density point in the assigned positive-current direction.

The current through an oriented surface is

\[I=\int_S\mathbf J\cdot d\mathbf A\]

Reversing the surface normal reverses the reported sign but does not change the physical motion.

Low-speed field of one moving point charge

For a point charge moving slowly compared with light, when retardation and acceleration radiation can be neglected, a useful approximation is

\[\mathbf B(\mathbf r) \approx\frac{\mu_0}{4\pi} \frac{q\,\mathbf v\times\mathbf R}{R^3}, \qquad \mathbf R=\mathbf r-\mathbf r_q\]

The vector \(\mathbf R\) points from the present source position in this quasistatic approximation to the observation point. Its magnitude is \(R=\lVert\mathbf R\rVert\). The vacuum permeability is

\[\mu_0\approx1.25663706212\times10^{-6} \ \mathrm{N/A^2}\]

Since the 2019 SI revision, this displayed decimal is measured rather than an exact defined value. For ordinary textbook calculations,

\(\mu_0/(4\pi)\approx10^{-7}\)

in SI units is usually sufficient.

The field magnitude is

\[B\approx\frac{\mu_0}{4\pi} \frac{|q|v\sin\theta}{R^2}\]

Here \(\theta\) is the angle between \(\mathbf v\) and \(\mathbf R\). The field is zero on the line of motion, greatest at right angles for fixed speed and distance, proportional to speed, and proportional to the signed source charge through its direction.

Cross-product direction

The cross product is perpendicular to both input vectors. In a right-handed coordinate system,

\[\hat{\mathbf x}\times\hat{\mathbf y} =\hat{\mathbf z}\]

Suppose a positive charge moves in the positive x direction and the observer lies in the positive y direction from the charge. Then

\(\mathbf v\times\mathbf R\)

points in the positive z direction. A negative charge reverses the field. Reversing the velocity also reverses the field.

The order of operands matters:

\[\mathbf R\times\mathbf v =-\mathbf v\times\mathbf R\]

import numpy as np

charge, mu_over_4pi = 1e-9, 1e-7
velocity = np.array([1000.0, 0.0, 0.0])
offset = np.array([0.0, 1.0, 0.0])
field = mu_over_4pi * charge * np.cross(velocity, offset) / np.linalg.norm(offset) ** 3
print("Slow-moving charge field (T):", field)

The exclusion radius in the Python is a modeling safeguard, not part of the point- charge formula. The ideal field becomes singular at the source position, where a finite- size charge distribution or a different physical model is required.

From moving charges to steady current

A wire contains many moving carriers. When their motion and density are steady on the timescale of interest, their contributions are summarized by the Biot–Savart law for a thin current element:

\[d\mathbf B =\frac{\mu_0}{4\pi} \frac{I\,d\boldsymbol\ell\times\hat{\mathbf R}}{R^2}\]

The oriented length element \(d\boldsymbol\ell\) points with conventional current, and \(\hat{\mathbf R}\) points from the source element to the observer. Integrating over the current path gives

\[\mathbf B(\mathbf r) =\frac{\mu_0 I}{4\pi} \int\frac{d\boldsymbol\ell\times\hat{\mathbf R}}{R^2}\]

This is a magnetostatic law. The same current must flow consistently through the modeled path, and source changes must be slow enough that propagation and radiation are not the question being studied.

Long straight wire

For an infinitely long straight wire carrying steady current \(I\), symmetry and integration give

\[B=\frac{\mu_0 I}{2\pi s}\]

The distance \(s\) is measured perpendicular to the wire. Field lines form circles centered on the wire. Their direction follows the cross product in the Biot–Savart law, often remembered with the right-hand rule: thumb with conventional current, curled fingers with magnetic field.

The formula predicts that doubling current doubles field, while doubling distance halves it. It does not apply at the wire axis or when the wire's finite radius and current distribution matter. Outside a long cylindrical wire with total current, the same inverse-distance result follows from Ampère's law.

Superposition

Magnetic fields add as vectors:

\[\mathbf B_{\mathrm{total}} =\sum_j\mathbf B_j\]

Add components, not magnitudes. Two equal fields in opposite directions cancel, while perpendicular contributions combine by vector addition. In a continuous source, the sum becomes an integral over current elements or current density.

sources = [
    (np.array([0.0, 0.0, 0.0]), 1e-9, np.array([1000.0, 0.0, 0.0])),
    (np.array([0.0, -1.0, 0.0]), 1e-9, np.array([1000.0, 0.0, 0.0])),
]
probe, total = np.array([0.0, 1.0, 0.0]), np.zeros(3)
for position, charge, velocity in sources:
    offset = probe - position
    total += 1e-7 * charge * np.cross(velocity, offset) / np.linalg.norm(offset) ** 3
print("Combined field (T):", total)

For a numerical line integral, halve the segment length and compare the vector field at the same observation points. Convergence of magnitude alone can conceal a direction error.

Magnetic field units and scale

The SI unit of magnetic field is the tesla. From the magnetic force law,

\[1\ \mathrm T =1\ \mathrm{N/(A\,m)} =1\ \mathrm{N\,s/(C\,m)}\]

Every result should include the observation position and coordinate basis. Reporting only “the field is two tesla” discards the direction that determines magnetic force and induction.

Causality and relativity

The moving-point-charge expression above uses a low-speed, quasistatic approximation. Electromagnetic changes do not propagate instantaneously. The exact fields of an arbitrarily moving point charge depend on the source at a retarded time, and acceleration produces radiation fields with a different distance dependence.

At relativistic speeds, the field is not obtained by simply inserting a large velocity into the low-speed formula. Electric and magnetic fields transform together between inertial frames. A source that is a pure charge density in one frame can appear as both charge and current density in another.

Verification and limits

Test basis directions before testing complicated geometry. A positive charge moving in positive x with an observer displaced in positive y must produce positive z field in the reference approximation. Reversing charge or velocity must negate the vector. Placing the observer on the motion line must make the cross product zero. For a long wire, field magnitude must scale linearly with current and inversely with distance.

The formulas assume ideal point charges or thin wires, a specified medium, and a quasistatic or magnetostatic regime. They omit finite source size, material magnetization, skin effect, source acceleration, retardation, radiation, and self-force. State which approximation is being used before treating a computed vector as a prediction.

A worked Python notebook

A positive one nanocoulomb charge moves at 1000 metres per second, and the probe lies perpendicular to its velocity. The question for this calculation is precise: how does magnetic field magnitude change when we change distance perpendicular to velocity, 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.

This is a low-speed, nonradiative approximation to a moving-charge field. 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 0.1 to 2 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 distance perpendicular to velocity, expressed in m. The output is magnetic field magnitude, expressed in T. 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.

\[f(q) = \frac{1.0 \cdot 10^{-13}}{q^{2}}\]

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 1 m, the expression gives 1e-13 T. 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("1e-7*1e-9*1000/q**2", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted magnetic field magnitude:", reference_output, "T")

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 1e-11 T. The midpoint, at 1.05 m, gives 9.07029e-14 T. The upper endpoint gives 2.5e-14 T. Read these together with the assumptions: they describe different choices of distance perpendicular to velocity 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([0.1, 1.05, 2.0])
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} m -> {output_value:.6g} T")
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 distance perpendicular to velocity; the vertical axis shows magnetic field magnitude. 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(0.1, 2.0, 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("Distance perpendicular to velocity (m)")
ax.set_ylabel("Magnetic field magnitude (T)")
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 magnetic field magnitude change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 m is -2e-13 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.

\[f'(q) = - \frac{2.0 \cdot 10^{-13}}{q^{3}}\]

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 = 1.0095
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, "T")
print("Full expression:", full_prediction, "T")
print("Approximation difference:", full_prediction - linear_prediction, "T")

Here the changed input is 1.0095 m. The tangent estimate is 9.81e-14 T, while the full expression gives 9.81267e-14 T. 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 = 1.9
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 distance perpendicular to velocity inside the stated interval, predict whether magnetic field magnitude 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: This is a low-speed, nonradiative approximation to a moving-charge field. 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 T: 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(2.0) - evaluate(0.1))
for points in [101, 201, 401]:
    grid = np.linspace(0.1, 2.0, 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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