Physics II · 120 h · Topic 8 of 15

Magnetostatics

There are no magnetic charges, so the magnetic field is not defined by the force on a charge at rest —that one is zero— but on a moving one. Everything comes from there, from the motor to the Hall sensor.

Lorentz force Biot–Savart Ampère's law Solenoid Hall effect

01The magnetic field is defined by what it does

There are no magnetic charges: there is no magnetic equivalent of an electron. That is why the magnetic field is not defined by the force on a charge at rest —that one is zero— but by the force on a charge in motion, and that force is perpendicular to everything.

\[ \vec{F} = q\,\vec{v}\times\vec{B} \qquad\qquad \vec{F} = I\,\vec{L}\times\vec{B} \] The Lorentz force, and its version for a conductor. Being perpendicular to the velocity, the magnetic force never does work: it changes the direction of motion but not the speed or the energy.

From the second expression comes every electric motor, and from the first the cathode-ray tube (CRT), the mass spectrometer and the Hall effect. And also a consequence worth keeping in mind: since there are no magnetic charges, the magnetic flux through any closed surface is always zero.

\[ \oint \vec{B}\cdot d\vec{A} = 0 \] The second of Maxwell's equations. All magnetic field lines close on themselves: they are neither born nor die anywhere, unlike electric ones.

02What produces a magnetic field

Currents. The Biot–Savart law gives the field of a current element, and Ampère's law does for magnetism what Gauss's law did for electrostatics: it turns an integration problem into a one-line calculation when there is symmetry.

\[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc} \ \mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m/A} \]

Configuration	Field	Where it appears
Infinite straight wire	\( B = \dfrac{\mu_0 I}{2\pi r} \)	Any cable; the field that couples into its neighbors
Center of a loop	\( B = \dfrac{\mu_0 I}{2R} \)	Deflection coils, loop antennas
Long solenoid	\( B = \mu_0 n I \)	Relays, electromagnets, inductors
Toroid	\( B = \dfrac{\mu_0 N I}{2\pi r} \)	Power-supply inductors: the field is confined

The third row has no distance: inside a long solenoid the field is uniform and depends only on the turns per meter. And the fourth explains why the inductors of switching power supplies are toroidal: in a toroid the field is enclosed inside the core and hardly radiates outward, which is what keeps the inductor from interfering with the rest of the board.

Lab · magnetic field of a current

Conductors perpendicular to the plane: the dot is current coming out toward you, the cross is going in. Tap anywhere to query the field there. With two conductors the force between them also appears.

Current = 10 A

03The force between conductors

Two parallel conductors attract each other if the currents flow in the same direction and repel if they flow the opposite way. Each one creates a field where the other is, and the other feels the Lorentz force.

\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \] With two wires one meter apart carrying one ampere each, the force is \( 2\times10^{-7} \) newton per meter. That number was the definition of the ampere until 2019, when the International System switched to defining it by fixing the charge of the electron.

The force is tiny with ordinary currents, but it grows with the square. In a switchboard busbar during a short circuit, tens of thousands of amperes flow, and the forces can reach the point of tearing off the supports: that is why busbars are calculated mechanically and not only by allowable current.

The pair of conductors that cancels out

If the two currents are equal and opposite and the conductors are very close together, their fields cancel almost completely at a distance. That is what a twisted pair does, and it is why the outgoing and return paths of a current must go together. The practical rule in board design is the same: the return path has to run right next to the outgoing one, because the area the two enclose is what determines how much field they generate and how much they pick up.

04The Hall effect

In a strip carrying current inside a perpendicular magnetic field, the Lorentz force pushes the carriers to one side until the electric field that builds up stops them. That transverse voltage is the Hall voltage:

\[ V_H = \frac{IB}{n q t} \] With \( n \) the carrier density and \( t \) the thickness. Since \( n \) is enormous in a metal and small in a semiconductor, Hall sensors are made of semiconductor: they give a voltage thousands of times greater.

The effect serves three purposes at once. It measures magnetic field, which is what a gaussmeter does; it measures current without contact, which is what a DC clamp meter does; and the sign of \( V_H \) reveals whether the carriers are negative or positive, which was the first experimental evidence that in a p-type semiconductor conduction is by holes.

05In the lab

Activity 1 · Field of a solenoid

Measure with a Hall probe the field on the axis of a solenoid for several currents. Verify the proportionality with \( I \) and compare the value with \( \mu_0 n I \). Then measure how it falls off beyond the end.

Activity 2 · The current balance

Hang a rigid conductor from a balance inside the air gap of a magnet and measure the force against the current. Obtain \( B \) from the slope and compare with the value measured with the probe.

Activity 3 · Magnetic shielding

Measure the field of a magnet through sheets of aluminum, steel and mu-metal. Explain why aluminum shields nothing and steel does, and relate it to the permeability of each material.

06Common mistakes

Believing the magnetic force does work. It is perpendicular to the velocity: it does not change the kinetic energy.
Looking for an isolated magnetic north pole. It does not exist; the closed flux always gives zero.
Confusing \( B \) with \( H \). \( B \) is the induction, \( H \) the excitation; in vacuum they differ by \( \mu_0 \).
Using the solenoid formula near the end. It holds for the interior of a long solenoid; at the mouth the field drops to half.
Separating the outgoing path from the return of a current. The enclosed area is what generates and picks up field.
Shielding a magnetic field with aluminum. For DC it does not work: you need a high-permeability material.

07Self-assessment

Why does the magnetic force do no work?

Because it is always perpendicular to the velocity: it changes the direction, not the speed.

What is the magnetic flux through a closed surface?

Zero, always. There are no magnetic charges and the lines close on themselves.

What does the field inside a long solenoid depend on?

Only on the turns per meter and the current: \( B = \mu_0 n I \). Not on the distance from the axis.

Why are power-supply inductors toroidal?

Because the toroid confines the field inside the core and hardly radiates to the rest of the board.

Why are Hall sensors made of semiconductor?

Because their carrier density is much lower than that of a metal, and the Hall voltage turns out thousands of times greater.

What force do two wires 1 m apart carrying 1 A exert on each other?

\( 2\times10^{-7} \) N per meter. It was the definition of the ampere until 2019.

08Further reading

Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). Magnetic field, Lorentz force, Biot–Savart and Ampère's law, with the Hall effect worked out.
Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed., Cambridge University Press, 2016 (Spanish edition, “Electricidad y magnetismo”). Magnetism deduced from electrostatics plus relativity, which is the most illuminating approach there is.
Henry W. Ott. Electromagnetic Compatibility Engineering. Wiley, 2009. Why the return path matters, the twisted pair and magnetic shielding in real equipment.