A Byte of PhysicsLogo

Magnetic fields due to currents

A steady electric current produces a magnetic field that circulates around the current path. Unlike the electric field of a positive point charge, the magnetic field does not simply point away from its source. Its direction depends on the oriented current path, the observer's position, and a cross product.

Two complementary tools are central. The Biot–Savart law adds contributions from current elements and works for many wire shapes. Ampère's law relates circulation of the field to enclosed current and becomes especially efficient when symmetry makes the field constant along a chosen path.

Biot–Savart law

For a thin wire carrying steady current \(I\), an oriented element \(d\boldsymbol\ell\) contributes

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

The vector \(\mathbf R\) points from the current element to the observation point, \(R=\lVert\mathbf R\rVert\), and \(\hat{\mathbf R}=\mathbf R/R\). The path element points with conventional current. Integrating gives

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

The cross product sets the direction. Reversing current reverses every contribution. Swapping the cross-product operands also reverses the field, but that swap would be an error rather than a new convention unless every equation were transformed consistently.

Reading the geometry

The magnitude of one small contribution is

\[dB=\frac{\mu_0}{4\pi} \frac{I\,d\ell\sin\theta}{R^2}\]

The angle \(\theta\) is between the current element and the displacement to the observer. A segment pointing directly toward or away from the observer contributes zero because \(\sin\theta=0\). A perpendicular segment gives the largest magnitude for fixed current, length, and distance.

Fields obey superposition:

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

Add vectors rather than magnitudes. Symmetry often cancels transverse components while reinforcing one shared component.

Infinite straight wire

For an infinitely long straight wire, the field magnitude at perpendicular distance \(s\) is

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

Field lines are circles centered on the wire. Point the right thumb with conventional current; curled fingers show the field direction. If positive current flows along positive z, then at a point on the positive x axis the field points in positive y.

In Cartesian components for a z-directed wire through the origin,

\[\mathbf B(x,y) =\frac{\mu_0 I}{2\pi(x^2+y^2)} \left(-y\,\hat{\mathbf x}+x\,\hat{\mathbf y}\right)\]

The denominator contains distance squared because the numerator contains one factor of distance. The resulting magnitude scales as \(1/s\). Doubling current doubles the field; doubling distance halves it.

Worked straight-wire value

At \(s=0.050\ \mathrm m\) from a long wire carrying \(I=4.0\ \mathrm A\),

\[B=\frac{(4\pi\times10^{-7}\ \mathrm{T\,m/A})(4.0\ \mathrm A)} {2\pi(0.050\ \mathrm m)} =1.6\times10^{-5}\ \mathrm T\]

This is \(16\ \mathrm{\mu T}\). At ten centimetres the field would be eight microtesla. The direction still requires the current orientation and observation side; the magnitude alone is incomplete.

Finite straight segment

An infinite-wire formula can be inaccurate near the end of a real wire. For a finite straight segment, the magnitude at perpendicular distance \(s\) can be written

\[B=\frac{\mu_0 I}{4\pi s} \left(\sin\theta_1+\sin\theta_2\right)\]

The endpoint angles are measured from the perpendicular drawn from the observer to the wire, using a common textbook geometry. When both endpoints recede toward infinity, each sine approaches one and the infinite-wire result is recovered. Because references may define these angles differently, always draw the angle convention before substituting.

Circular current loop

For a circular loop of radius \(a\) carrying current \(I\), transverse contributions cancel at the center and axial contributions add. The central field magnitude is

\[B_{\mathrm{center}}=\frac{\mu_0 I}{2a}\]

For \(N\) closely packed turns carrying the same current, multiply by \(N\). Along the symmetry axis a distance \(z\) from the center,

\[B_z(z)=\frac{\mu_0 I a^2} {2(a^2+z^2)^{3/2}}\]

At \(z=0\), this reduces to the central formula. Far from the loop, the field falls approximately as inverse distance cubed, characteristic of a magnetic dipole.

Ampère's law

For steady currents, Ampère's law is

\[\oint_C\mathbf B\cdot d\boldsymbol\ell =\mu_0 I_{\mathrm{enclosed}}\]

The closed integration path is called an Amperian loop. The sign of enclosed current is set by the path orientation and the corresponding right-hand-rule surface normal. Currents outside the chosen surface can affect the local field, but their net contribution to this circulation is zero in magnetostatics.

Ampère's law is always a true integral constraint in its magnetostatic domain, but it only gives the field directly when symmetry tells us the field direction and makes its magnitude constant or zero on path segments. For a circular path around an infinite straight wire,

\[B(2\pi s)=\mu_0 I, \qquad B=\frac{\mu_0 I}{2\pi s}\]

Solenoids and toroids

An ideal long solenoid has \(n=N/L\) and carries current \(I\). Far from its ends, the internal field is approximately uniform:

\[B\approx\mu_0 nI\]

The exterior field is small compared with the interior field in the long-solenoid approximation. Near the ends, the field bends outward and the approximation becomes less accurate.

For an ideal toroid with \(N\) turns, Ampère's law gives within the winding region

\[B(s)=\frac{\mu_0 NI}{2\pi s}\]

The field circles the toroid's central axis and varies with radial distance. In the ideal tightly wound model, the field outside is approximately zero. A magnetic core replaces the vacuum response with a material relation that may be nonlinear and history dependent.

Numerical Biot–Savart integration

For a curved or finite path without a convenient analytic result, divide the wire into short oriented segments and add their vector contributions.

import numpy as np

radius, current, segments = 0.1, 2.0, 1000
angles = np.linspace(0, 2 * np.pi, segments + 1)
points = np.column_stack(
    [radius * np.cos(angles), radius * np.sin(angles), np.zeros_like(angles)]
)
probe, field = np.array([0.0, 0.0, 0.2]), np.zeros(3)
for start, end in zip(points[:-1], points[1:]):
    offset = probe - (start + end) / 2
    field += (
        1e-7 * current * np.cross(end - start, offset) / np.linalg.norm(offset) ** 3
    )
exact_z = 2 * np.pi * 1e-7 * current * radius**2 / (radius**2 + probe[2] ** 2) ** 1.5
print("Numerical field (T):", field)
print("Analytic axial field (T):", exact_z)

The midpoint sum approximates the line integral. Halve the maximum segment length and compare all vector components at the same observation point. Report the minimum distance to the wire and never hide samples inside a singular thin-wire model by returning zero.

Force between parallel currents

The field of one wire acts on current in another. For two long parallel wires separated by distance \(s\), the force per unit length is

\[\frac{F}{L}=\frac{\mu_0 I_1I_2}{2\pi s}\]

Currents in the same direction attract; currents in opposite directions repel. This direction follows by combining the circular field of the first wire with the magnetic force \(I\boldsymbol\ell\times\mathbf B\) on the second.

Verification and limits

Test directions with basis vectors before complex wire paths. Reverse current and confirm that every field component reverses. For a long wire, confirm linear scaling with current and inverse scaling with distance. For a loop, confirm the axial expression reduces to the central value. For numerical integration, check convergence under segment refinement.

Biot–Savart and the simple form of Ampère's law here assume steady currents. Time-varying fields require Maxwell's displacement-current term and causal electromagnetic solutions. Ideal wires omit finite cross-section, skin effect, contacts, return paths, magnetic materials, and nearby conductors. Near the wire or its ends, use the actual geometry and current distribution rather than extending an infinite thin-wire result.

A worked Python notebook

Evaluate the axial field of a circular loop of radius 0.1 metres carrying two amperes. The question for this calculation is precise: how does axial magnetic field change when we change axial distance from loop centre, 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 current is steady, the wire is thin, and the probe lies on the symmetry axis. 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 0.5 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 axial distance from loop centre, expressed in m. The output is axial magnetic field, 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.

\[f(q) = \frac{4.0 \cdot 10^{-9} \pi}{\left(q^{2} + 0.01\right)^{\frac{3}{2}}}\]

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 0.2 m, the expression gives 1.12397e-06 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("2*pi*1e-7*2*0.1**2/(0.1**2+q**2)**(3/2)", locals={"q": q})
reference_input = 0.2
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted axial magnetic field:", 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 1.25664e-05 T. The midpoint, at 0.25 m, gives 6.43729e-07 T. The upper endpoint gives 9.47872e-08 T. Read these together with the assumptions: they describe different choices of axial distance from loop centre 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, 0.25, 0.5])
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 axial distance from loop centre; the vertical axis shows axial magnetic 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, 0.5, 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("Axial distance from loop centre (m)")
ax.set_ylabel("Axial magnetic field (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 axial magnetic field change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.2 m is -1.34876e-05 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.

\[f'(q) = - \frac{1.2 \cdot 10^{-8} \pi q}{\left(q^{2} + 0.01\right)^{\frac{5}{2}}}\]

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 = 0.2025
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 0.2025 m. The tangent estimate is 1.09025e-06 T, while the full expression gives 1.09087e-06 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 = 0.5
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 axial distance from loop centre inside the stated interval, predict whether axial magnetic 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 current is steady, the wire is thin, and the probe lies on the symmetry axis. 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(0.5) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.5, 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

Share to: