Magnetostatics §

The idea, step by step.

1. A current element is the magnetic analogue of a point charge. Just as a point charge $q$ produces an electric field $\mathbf{E}\propto q\,\hat{\mathbf{r}}/r^2$ (Coulomb’s law, Electrostatics), a tiny piece of wire carrying current $I$ produces a magnetic field. The Biot–Savart law is the magnetic “Coulomb’s law”: it gives the field of the smallest source element, and everything else is built by superposition.

2. The cross product sets the direction. The factor $d\boldsymbol{\ell}\times\hat{\mathbf{r}}$ means the field is perpendicular to both the current direction and the line to the field point. Curl the fingers of your right hand in the direction of the current; your thumb points along $d\boldsymbol{\ell}$ and your curled fingers show how $\mathbf{B}$ circles the wire. This is why the field of a straight wire is a set of concentric circles around it.

3. Integrate to get the field of a real wire. For a long straight wire, every element contributes a field tangent to a circle of radius $r$ around the wire, and all contributions point the same way. Integrating around the whole wire gives the classic result

The field falls as $1/r$ (not $1/r^2$ as in electrostatics) because a wire is an extended source, not a point.

4. Authoritative source. See the Biot–Savart law and Griffiths, Introduction to Electrodynamics, Ch. 5.

The idea, step by step.

1. The circulation counts the enclosed current. The line integral $\oint \mathbf{B}\cdot d\boldsymbol{\ell}$ sums the component of the field along a closed path. Ampère’s law says this circulation depends only on how much current passes through the surface bounded by the path — currents outside the loop contribute nothing to the net circulation. It is a “current counter,” exactly as Gauss’s law is a “charge counter.”

2. Why the field of a straight wire satisfies it. Take a circular loop of radius $r$ around a long straight wire. By the right-hand rule $\mathbf{B}$ is everywhere tangent to the loop and has constant magnitude $B$, so the integral is $B(2\pi r)$. The enclosed current is $I$, so $B(2\pi r)=\mu_0 I$, recovering $B=\mu_0 I/(2\pi r)$ from Section 1 — but without doing the Biot–Savart integral.

3. Symmetry turns it into a calculator. Choose an Ampèrian loop on which $\mathbf{B}$ is either constant and parallel to $d\boldsymbol{\ell}$ or perpendicular to it (so it contributes nothing):

   • Infinite straight wire (cylindrical symmetry): a coaxial circle gives $B = \mu_0 I/(2\pi r)$.
   • Long solenoid (a coil of $n$ turns per unit length): a rectangular loop straddling the wall gives a uniform interior field $B = \mu_0 n I$ and (ideally) zero outside.
   • Toroid (a coil wound on a ring, $N$ total turns): a circular loop of radius $r$ inside the core gives $B = \mu_0 N I/(2\pi r)$, and zero outside.

4. Authoritative source. Ampère’s law is one of Maxwell’s equations; see Maxwell (1865) and Griffiths, Introduction to Electrodynamics, Ch. 5.

The idea, step by step.

1. No monopoles force a vector potential. In electrostatics the field is conservative ($\nabla\times\mathbf{E}=0$), so it is the gradient of a scalar $V$. In magnetostatics the field is solenoidal ($\nabla\cdot\mathbf{B}=0$, no magnetic monopoles), so it is the curl of a vector $\mathbf{A}$. The two static regimes are exact mirrors: a scalar potential whose gradient gives $\mathbf{E}$, and a vector potential whose curl gives $\mathbf{B}$.

2. The potential is a sum of scalars, then differentiate. For a steady current the vector potential at a point is just a scalar sum over the current elements, $\mathbf{A} = (\mu_0/4\pi)\int I\,d\boldsymbol{\ell}'/r'$ — far easier to add than the vector Biot–Savart field. Once you have $\mathbf{A}$, the field is its curl. This “sum, then differentiate” strategy parallels the electrostatic one and is the workhorse for more complicated geometries.

3. Gauge freedom. $\mathbf{A}$ is not unique: adding the gradient of any scalar, $\mathbf{A}\to\mathbf{A}+\nabla\chi$, leaves $\mathbf{B}=\nabla\times\mathbf{A}$ unchanged. Only $\mathbf{B}$ (and, in quantum mechanics, the line integral of $\mathbf{A}$) is physically observable — the content of the Aharonov–Bohm effect.

4. Authoritative source. See Griffiths, Introduction to Electrodynamics, Ch. 5 (vector potential), and Maxwell (1865).

The idea, step by step.

1. The magnetic force is always sideways. The cross product $\mathbf{v}\times\mathbf{B}$ means the force is perpendicular to both the velocity and the field, so it changes the direction of a moving charge but never its speed. A magnetic field does no work on a charge; it only bends the trajectory into a circle or helix. This is the origin of the circular field lines of Section 1 — a moving charge’s own field circles its path.

2. A current in a field feels a force. The charges drifting through a wire each feel $q\mathbf{v}\times\mathbf{B}$; summed over the wire this is $d\mathbf{F}=I\,d\boldsymbol{\ell}\times\mathbf{B}$. This is the principle behind the electric motor (a current-carrying coil in a field experiences a torque) and the Hall effect.

3. Parallel wires attract or repel. Wire 1 creates a circular field $B_1 = \mu_0 I_1/(2\pi d)$ at the location of wire 2. The current in wire 2 then feels $F/L = I_2 B_1 = \mu_0 I_1 I_2/(2\pi d)$. The right-hand rule shows the force pulls the wires together when the currents flow the same way and pushes them apart when they oppose. This force per unit length is in fact the historical definition of the ampere.

4. Authoritative source. See Griffiths, Introduction to Electrodynamics, Ch. 5 (force on currents), and Maxwell (1865).

The idea, step by step.

1. Why a second field, H? In vacuum $\mathbf{B}=\mu_0\mathbf{H}$ and the two are proportional, so either suffices. In a material, however, the total current that sources $\mathbf{B}$ has two parts: the free current you drive through a wire, and the bound current from the material’s microscopic magnetic dipoles aligning. The field $\mathbf{H}$ is defined so that Ampère’s law counts only the free current, $\nabla\times\mathbf{H}=\mathbf{J}_{\text{free}}$, leaving the material’s response in the relation $\mathbf{B}=\mu\mathbf{H}$. This is the magnetic twin of the $\mathbf{D}$/$\mathbf{E}$ split in Electromagnetic Media.

2. Permeability measures how easily a material “conducts” magnetic field. A vacuum has $\mu_r=1$. A diamagnet has $\mu_r$ just below 1 (it weakly opposes the field); a paramagnet just above 1 (it weakly enhances it); a ferromagnet (iron, nickel, cobalt) has a large, field-dependent $\mu_r$ that can reach thousands, which is why an iron core multiplies the field of a solenoid by a large factor.

3. Permanent magnets are steady currents in disguise. A permanent magnet’s field comes from the alignment of atomic magnetic moments (electron spin and orbital motion). Macroscopically this is equivalent to a bound surface current flowing around the magnet, so the field of a bar magnet is exactly the field of a steady current loop — the dipole field of Section 6. Magnetostatics therefore describes a permanent magnet without any time dependence: the “current” is frozen into the material.

4. Authoritative source. See Electromagnetic Media for the full constitutive treatment, and Griffiths, Introduction to Electrodynamics, Ch. 5 (magnetic materials).

The idea, step by step.

1. A loop is the simplest magnetic dipole. A single current loop has no net “magnetic charge,” but its field is concentrated on one face and returns on the other — just like the north and south poles of a bar magnet. The strength and orientation of this dipole are captured by the single vector $\mathbf{m}=IA\hat{\mathbf{n}}$. More loops (a coil of $N$ turns) give $\mathbf{m}=NIA\hat{\mathbf{n}}$.

2. The far field falls as 1/r³. Close to the loop the field is complicated, but far away only the dipole term survives and it decays as $1/r^3$ — faster than the $1/r$ of a wire or the $1/r^2$ of a point charge. On the axis the field is $B = (\mu_0/4\pi)(2m/r^3)$, pointing along $\mathbf{m}$.

3. Dipoles are everywhere. A bar magnet’s field is a dipole field (Section 5); the Earth’s magnetic field is, to first order, a dipole field from its core; and the magnetic moment of an electron is the quantum dipole that makes ferromagnetism possible. The dipole is the universal building block of magnetostatic fields.

4. Authoritative source. See Griffiths, Introduction to Electrodynamics, Ch. 5 (magnetic dipole field), and Maxwell (1865).

Problem. A long solenoid of radius $R = 0.04\,\text{m}$ has $n = 2000$ turns per metre and carries a current $I = 1.5\,\text{A}$.

1. What is the magnetic field at the centre of the solenoid?
2. What is the magnetic flux through a circular cross-section of the solenoid (radius $R$)?
3. If the current is doubled, by what factor does the field energy stored per unit length change? (The magnetic energy density is $u = B^2/(2\mu_0)$.)