Physics · Electricity and magnetism · Grade 8

Magnetic field

What turns a compass needle — the Earth, a magnet or a current in a wire?

The situation

Put a tourist compass on a table in Tashkent and its needle points north. Bring a magnet next to it and the needle turns towards the magnet; run a current-carrying wire over it and it turns sideways. In 1820 Oersted saw exactly this, and discovered that an electric current produces a magnetic field. What turns the needle, how do we measure the field, and how strong is the Earth's field?

Topic

Magnetic field. Compass, Earth's magnetic field and Ørsted's experiment

Key ideas

Magnetic flux density B
Oersted's experiment
Biot–Savart law

Formulas

B = \frac{F_{m a x}}{I Δ l}, [B] = 1 Tl = 1 \frac{N}{A \cdot m}

The magnitude of the magnetic flux density vector. Its direction is the direction pointed to by the N end of a compass needle.

d \vec{B} = \frac{μ_{0}}{4 π} \frac{I d \vec{l} \times \vec{r}}{r^{3}}, \vec{B} = \sum_{i} {\vec{B}}_{i}

The Biot–Savart–Laplace law and the principle of superposition: the field is the vector sum of the fields of all the current elements.

B_{w i r e} = \frac{μ_{0} I}{2 π r}, B_{l o o p} = \frac{μ_{0} N I}{2 R}, B_{solenoid} = μ_{0} n I

The field of a straight wire, at the centre of a loop and inside a long solenoid; $μ_{0} = 4 π \cdot 10^{- 7}$ T·m/A, $n = N / l$ .

\oint_{L} \vec{B} \cdot d \vec{l} = μ_{0} \sum I_{e n c}, \oint_{S} \vec{B} \cdot d \vec{S} = 0

Ampère's circuital law and Gauss's law for magnetism: there is no magnetic “charge”, and field lines are closed.

\frac{F}{l} = \frac{μ_{0} I_{1} I_{2}}{2 π d}, \tan α = \frac{μ_{0} N I}{2 R B_{h o r}}

The force between parallel currents and the formula for the tangent galvanometer.

Theory

The permanent magnet, its poles and field lines

A magnet has two poles, N (north) and S (south); like poles repel and unlike poles attract. The field is described by the magnetic flux density vector $\vec{B}$ (unit: tesla, T): its direction is the direction pointed to by the N end of a compass needle. Field lines are tangent to $\vec{B}$ at every point; where they are dense, the field is strong. Iron filings form chains along the field lines — but only where the field is ≳ 0.1 mT; farther away they stay disordered. Inside the magnet the lines continue from S to N and so close up: there is no isolated pole, so the magnetic flux through any closed surface is $Φ = \oint \vec{B} \cdot d \vec{S} = 0$ (Gauss's law). For a flat surface $Φ = B S \cos α$ , measured in webers (Wb).

Earth's magnetic field

The Earth is a giant magnet: its field resembles that of a dipole tilted by ≈ 9° to the axis of rotation. The magnetic pole near the geographic North Pole is in fact a south (S) magnetic pole — it attracts the N end of a compass. The needle deviates from geographic north by an angle $D$ (declination), and a freely suspended needle dips below the horizontal by an angle $I$ (inclination, or dip). In the dipole model $\tan I = 2 \tan φ_{m}$ ( $φ_{m}$ is the geomagnetic latitude). For Tashkent (IGRF-14 model, 2026, approximately): $D \approx 5.7 °$ east, $I \approx 62 °$ , $B \approx 54$ µT, horizontal component $B_{h o r} \approx 25.5$ µT. The dipole model gives $I \approx 53 °$ for Tashkent — the difference comes from the non-dipole part of the field.

Oersted's experiment, the Biot–Savart–Laplace law

A magnetic field appears around a wire carrying current — a compass needle turns. Each current element $I d \vec{l}$ gives at a point the field $d \vec{B} = \frac{μ_{0}}{4 π} \frac{I d \vec{l} \times \vec{r}}{r^{3}}$ ; the field of the whole conductor is their sum (the principle of superposition). Every field in this scene is calculated in exactly this way: the loop and the solenoid are divided into small straight segments. The results: for a straight wire $B = μ_{0} I / (2 π r)$ , with lines that are concentric circles (the right-hand screw rule: if a screw is turned so that it advances in the direction of the current, the direction in which its handle turns shows the direction of the field lines); at the centre of a loop $B = μ_{0} N I / (2 R)$ . A charge moving steadily also produces such a field: $\vec{B} = \frac{μ_{0}}{4 π} \frac{q \vec{v} \times \vec{r}}{r^{3}}$ (at low speeds).

Ampère's circuital law, the solenoid and the toroid

The circulation of $\vec{B}$ around a closed loop depends on the sum of the currents enclosed by the loop: $\oint \vec{B} \cdot d \vec{l} = μ_{0} \sum I$ . In “Straight wire” mode switch on the loop: if the wire is inside the loop the sum is $μ_{0} I$ , and outside it is zero; the shape and size of the loop do not matter. Because the circulation is not zero, the field is rotational (not conservative); it can be written through the vector potential: $\vec{B} = curl \vec{A}$ . Inside a long solenoid the field is almost uniform: $B = μ_{0} n I$ , and at its ends it is about half as large. A solenoid bent into a ring — a toroid — has $B = μ_{0} N I / (2 π r)$ inside and almost no field outside. When the current in a coil changes, self-induction occurs — see it in the “Self-induction” experiment.

The interaction of currents, the Ampère and Lorentz forces

A magnetic field acts on a current-carrying conductor with the Ampère force $F = I l B \sin α$ , and on a moving charge with the Lorentz force $F = q v B \sin α$ (the motion of charges in a field is covered in the “Aurora and the Lorentz force” experiment and the accelerator experiments). That is why two parallel currents interact: $F / l = μ_{0} I_{1} I_{2} / (2 π d)$ ; currents in the same direction attract and opposite ones repel. Until 2019 the ampere was defined in this way: for two long wires 1 m apart, a current of 1 A gives a force of $2 \cdot 10^{- 7}$ N per metre. Our teslameter is a Hall sensor: the Lorentz force deflects the charges in it sideways and a voltage $U_{H} \propto I B$ appears. The sensor is zeroed relative to the Earth's field — it shows only the field of a magnet or a current.

Laboratory: measuring B and the tangent galvanometer

Determining the magnetic flux density: in the “Straight wire”, “Loop” and “Solenoid” modes move the teslameter, measure $B$ and compare it with the formulas (the current is set by the source voltage and the rheostat resistance through Ohm's law: $I = U / R$ ). Tangent galvanometer: a compass stands at the centre of a vertical loop of $N$ turns (radius $R = 10$ cm). The plane of the loop is placed along the magnetic meridian, so that the field of the loop, $B = μ_{0} N I / (2 R)$ , is perpendicular to the horizontal component of the Earth's field and the needle turns through an angle $\tan α = B / B_{h o r}$ . Measure $α$ at several currents, and find $B_{h o r} = μ_{0} / (2 R \cdot k)$ from the slope of the graph of $\tan α$ against $N I$ ; the constant of the tangent galvanometer is $C = I / \tan α = 2 R B_{h o r} / (μ_{0} N)$ . To remove the zero error, reverse the current and take the mean of the two angles; for accuracy it is best for $α$ to lie between 20° and 70°.

Did you know?

In 1820 the Danish physicist Hans Christian Oersted observed the deflection of a compass needle beside a current-carrying wire and was the first to announce that an electric current produces a magnetic field.
Until 2019 the ampere was defined through the interaction of two parallel currents ( $2 \cdot 10^{- 7}$ N/m). Since 20 May 2019 it has been defined through the exact value of the elementary charge, $e = 1.602176634 \cdot 10^{- 19}$ C.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Capacitor and capacitance — Grade 8
Circuits and Kirchhoff's rules — Grade 8
Electric current in metals — Grade 8
Electrolysis, Faraday's laws — Grade 8
Electric current in vacuum and gases — Grade 8
Electromagnet and relay — Grade 8
Shake torch — Grade 10
Self-induction — Grade 10
Magnetic properties of matter — Grade 10
Plasma — the fourth state — Grade 10