Chapter 5

Biot-Savart Law & Ampere's Law

Magnetic forces, the Biot-Savart law, Ampere's law, and the divergence and curl of B.

5.1 Magnetic Force & Lorentz Force

A charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$experiences the magnetic Lorentz force:

$$\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$$

The magnetic force $q\mathbf{v} \times \mathbf{B}$ is always perpendicular to $\mathbf{v}$, so it does no work. The force per unit length between two parallel wires carrying currents$I_1$ and $I_2$ separated by distance $d$ is:

$$\frac{F}{\ell} = \frac{\mu_0}{2\pi}\frac{I_1 I_2}{d}$$

5.2 The Biot-Savart Law

The magnetic field produced by a steady current $I$ in a wire element $d\boldsymbol{\ell}'$ at position $\mathbf{r}'$ is given by the Biot-Savart law:

$$\boxed{d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I\,d\boldsymbol{\ell}' \times \hat{\mathscr{r}}}{\mathscr{r}^2}}$$

$\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{r}'$, $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$

For a general current distribution:

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}') \times \hat{\mathscr{r}}}{\mathscr{r}^2}\,d\tau'$$

5.2.1 Magnetic Field on Axis of a Circular Loop

For a circular loop of radius $R$ carrying current $I$, at a point on the axis distance $z$ from the center:

By symmetry, only the axial component $B_z$ survives. Each element $Rd\phi$contributes $dB_z = (\mu_0 I R\,d\phi)/(4\pi\mathscr{r}^2) \cdot R/\mathscr{r}$ where$\mathscr{r} = \sqrt{R^2 + z^2}$. Integrating over $2\pi$:

$$\boxed{B_z = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}}$$

At center ($z=0$): $B = \mu_0 I / (2R)$

5.3 Ampere's Law

Analogous to Gauss's law, Ampere's law exploits symmetry to find $\mathbf{B}$ without integrating the Biot-Savart law:

$$\boxed{\oint_C \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_{\rm enc}} \qquad \Longleftrightarrow \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$

Infinite solenoid (n turns/m)

$$B = \mu_0 n I \quad \text{(inside)}, \quad B = 0 \quad \text{(outside)}$$

Infinite wire (radius $s$)

$$\mathbf{B} = \frac{\mu_0 I}{2\pi s}\,\hat{\phi} \quad (s > R)$$

5.3.1 Divergence and Curl of B

$$\nabla \cdot \mathbf{B} = 0 \qquad \text{(no magnetic monopoles)}$$$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} \qquad \text{(Ampere's law, static case)}$$

$\nabla \cdot \mathbf{B} = 0$ is the second of Maxwell's four equations. It states there are no magnetic charges (monopoles).

Simulation: Biot-Savart for a Circular Loop

Computes $\mathbf{B}$ by direct numerical summation of the Biot-Savart law, compares with the analytic on-axis formula, and maps the 2D field in the $xz$-plane.

Biot-Savart: Circular Current Loop

Numerical Biot-Savart integration for a circular loop, compared against the analytic on-axis result.

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Biot-Savart: magnetic field of a circular loop
# and comparison with on-axis analytic formula
# ──────────────────────────────────────────────
mu0 = 4 * np.pi * 1e-7   # permeability of free space [T·m/A]
I = 1.0                   # current [A]
R = 1.0                   # loop radius [m]
N_seg = 500               # number of wire segments

# Discretize the circular loop in the xy-plane
phi = np.linspace(0, 2*np.pi, N_seg, endpoint=False)
dphi = phi[1] - phi[0]
# Wire element positions
rx = R * np.cos(phi)
ry = R * np.sin(phi)
rz = np.zeros(N_seg)
# Wire element dl
dlx = -R * np.sin(phi) * dphi
dly =  R * np.cos(phi) * dphi
dlz = np.zeros(N_seg)

def B_field(x, y, z):
    """Compute B at point (x, y, z) by Biot-Savart summation."""
    Bx = By = Bz = 0.0
    for i in range(N_seg):
        dx = x - rx[i]; dy = y - ry[i]; dz = z - rz[i]
        r = np.sqrt(dx**2 + dy**2 + dz**2)
        if r < 1e-10:
            continue
        # dl × r_hat / r²
        r3 = r**3
        Bx += (dly[i]*dz - dlz[i]*dy) / r3
        By += (dlz[i]*dx - dlx[i]*dz) / r3
        Bz += (dlx[i]*dy - dly[i]*dx) / r3
    return mu0*I/(4*np.pi) * np.array([Bx, By, Bz])

# ── On-axis field (analytic) ──
z_axis = np.linspace(-3, 3, 200)
Bz_analytic = mu0 * I * R**2 / (2 * (R**2 + z_axis**2)**1.5)

# ── Numerical on-axis ──
Bz_num = np.array([B_field(0, 0, z)[2] for z in z_axis])

# ── 2D field map in xz-plane (y=0) ──
x1d = np.linspace(-2.5, 2.5, 30)
z1d = np.linspace(-2.5, 2.5, 30)
Xg, Zg = np.meshgrid(x1d, z1d)
BXg = np.zeros_like(Xg); BZg = np.zeros_like(Zg)
for xi in range(len(x1d)):
    for zi in range(len(z1d)):
        b = B_field(x1d[xi], 0, z1d[zi])
        BXg[zi, xi] = b[0]
        BZg[zi, xi] = b[2]

fig, axes = plt.subplots(1, 3, figsize=(16, 5.5))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

# On-axis comparison
axes[0].plot(z_axis, Bz_analytic*1e6, color='#60a5fa', lw=3, label='Analytic')
axes[0].plot(z_axis, Bz_num*1e6,     'r--', lw=1.8, label='Biot-Savart (numerical)')
axes[0].axvline(-R, color='gray', lw=1, linestyle=':')
axes[0].axvline( R, color='gray', lw=1, linestyle=':', label='Loop rim at ±R')
axes[0].set_xlabel('z [m]', color='#94a3b8'); axes[0].set_ylabel(r'$B_z$ [μT]', color='#94a3b8')
axes[0].set_title('On-axis B_z: analytic vs Biot-Savart', color='white', fontsize=11)
axes[0].tick_params(colors='#94a3b8')
axes[0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0].grid(alpha=0.2, color='#334155')

# 2D streamlines
B_mag = np.sqrt(BXg**2 + BZg**2)
axes[1].streamplot(Xg, Zg, BXg, BZg, color=np.log10(B_mag+1e-10),
                   cmap='plasma', density=1.5, linewidth=1.1, arrowsize=1.2)
axes[1].plot([-R, R], [0, 0], 'o', color='#60a5fa', ms=10, label='Loop (cross-section)')
axes[1].set_xlabel('x [m]', color='#94a3b8'); axes[1].set_ylabel('z [m]', color='#94a3b8')
axes[1].set_title('B-field in xz-plane (Biot-Savart)', color='white', fontsize=11)
axes[1].set_aspect('equal')
axes[1].tick_params(colors='#94a3b8')
axes[1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)

# Relative error on axis
rel_err = np.abs(Bz_num - Bz_analytic) / (np.abs(Bz_analytic) + 1e-20)
axes[2].semilogy(z_axis, rel_err + 1e-10, color='#f59e0b', lw=2)
axes[2].set_xlabel('z [m]', color='#94a3b8'); axes[2].set_ylabel('Relative error', color='#94a3b8')
axes[2].set_title('Numerical vs Analytic Error', color='white', fontsize=11)
axes[2].tick_params(colors='#94a3b8')
axes[2].grid(alpha=0.2, color='#334155')
axes[2].set_ylim(1e-5, 1)

plt.tight_layout()
plt.savefig('biot_savart.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
print(f"Max relative error (on-axis): {rel_err.max():.2e}")
print(f"B_z at center (z=0): numerical={Bz_num[100]*1e6:.4f} μT, analytic={Bz_analytic[100]*1e6:.4f} μT")

Click Run to execute the Python code

First run will download Python environment (~15MB)

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