magnetic induction and superposition of magnetic fields

Magnetic induction $\vec{B}$ is the vector force characteristic of a magnetic field: it determines the force acting on a moving charge or on a current element, and it is measured in teslas (T). The direction of $\vec{B}$ for each source is found by the right-hand rule (grip rule). Simple sources have ready formulas: a long straight wire $B = \frac{\mu_0 I}{2\pi r}$, the centre of a circular loop $B = \frac{\mu_0 I}{2R}$, a long solenoid $B = \mu_0 n I$. Fields of different sources do not distort one another: the resultant induction is the vector sum $\vec{B} = \sum_i \vec{B}_i$, that is, the superposition principle holds, and the field of each source is computed as if the others were absent. Therefore every problem is solved in steps: first find the magnitude and direction of each contribution, then add them as vectors. The currents themselves are found from Ohm's law, and since the resistance of a metal grows with temperature, heating by the current can change the current and the field it creates.

A long straight wire carrying a current $I = 5.0$ A is perpendicular to the plane of the drawing, the current pointing towards the reader (out of the page). Find the magnitude of the magnetic induction at a point $r = 2.0$ cm from the wire and state the direction of $\vec{B}$.

Two long parallel wires are a distance $d = 0.20$ m apart. Currents $I_1 = 3.0$ A and $I_2 = 5.0$ A flow in the same direction, perpendicular to the plane of the drawing, towards the reader. Find the magnetic induction at the point $M$ midway between the wires.

Two long straight wires cross at right angles and are insulated from each other. Wire 1 lies along the $x$-axis with current $I_1 = 4.0$ A in the $+x$ direction. Wire 2 lies along the $z$-axis with current $I_2 = 6.0$ A in the $+z$ direction. Point $P$ has coordinates $(0;\,0.10;\,0)$ m. Find the magnitude and direction of the resultant magnetic induction at $P$.

Self-check plan in a simulation (safe, no real circuits).

Open a browser simulation of the magnetic field of currents (for example, PhET "Faraday's Electromagnetic Lab", the electromagnet tab, or any 2D simulator of the field of current-carrying conductors).

1. Place two long straight conductors perpendicular to the screen, set $I_1 = 3$ A and $I_2 = 5$ A in the same direction, and mark a point $P$ midway between them.
2. Switch on only the first conductor and record the magnitude and direction of $\vec{B}_1$ at $P$ (use the simulator's probe).
3. Switch off the first and switch on only the second conductor, and record $\vec{B}_2$.
4. Switch on both and compare the observed vector with the prediction $\vec{B} = \vec{B}_1 + \vec{B}_2$; calculate the expected result with the formulas from the worked examples.
5. Reverse one current and find the position of the point where the field becomes zero; check that there $B_1 = B_2$.
6. Separate the conductors so that the contributions become perpendicular, and verify that the resultant magnitude equals $\sqrt{B_1^2 + B_2^2}$, not $B_1 + B_2$.

Safety: carry out all observations in the simulation. Do not measure magnetic fields near household mains wiring and do not connect conductors to the mains; in a real experiment use only a battery up to 1.5 V with a current-limiting resistor.

Show that the resultant magnetic induction at a point is the vector sum of the contributions of individual sources, not the sum of their magnitudes, so the direction of every contribution must be constructed before adding.

Diagram (view perpendicular to the conductors): two long conductors with currents $I_1$ and $I_2$ and a point $P$; at $P$ the vectors $\vec{B}_1$ and $\vec{B}_2$ are drawn with labelled magnitudes, next to them the vector parallelogram is constructed, its diagonal is labelled $\vec{B} = \vec{B}_1 + \vec{B}_2$, the angle $\alpha$ between the contributions is shown, and the antiparallel case with $B = |B_2 - B_1|$ is marked separately.

A disc magnet is fixed to a bicycle spoke. When the spoke passes the reed switch, the distance from the magnet's centre to the switch is $r \approx 0{,}01$ m and the magnet's field there is $B_{\text{m}} \approx 0{,}02$ T. The Earth's magnetic induction at the same place is $B_{\text{E}} \approx 5\cdot10^{-5}$ T. A wheel of diameter $0{,}66$ m at $6{,}0$ m/s turns at $2{,}9$ rev/s, so the magnet passes the switch $2{,}9$ times per second.

The switch triggers when the induction vector at its location exceeds a threshold. The total field here is the vector sum $\vec{B} = \vec{B}_{\text{m}} + \vec{B}_{\text{E}}$: $0{,}02$ T against $5\cdot10^{-5}$ T, i.e. the magnet's field is 400 times larger, so the magnitude and direction of the sum nearly coincide with the magnet's field and the Earth's contribution does not change the switching.

A car's starter cable carries a current $I = 120$ A and runs at a distance $r = 0{,}30$ m from the compass on the dashboard. The field of a straight wire at the compass is $B = \frac{\mu_0 I}{2\pi r} = \frac{2\cdot10^{-7}\cdot120}{0{,}30} = 8\cdot10^{-5}$ T $= 80$ µT. The horizontal component of the Earth's field at mid-latitudes is $B_{\text{E}} \approx 2\cdot10^{-5}$ T $= 20$ µT. The current flows for $1{,}0$ s while the engine starts.

The compass needle settles along the total field $\vec{B}_{\Sigma} = \vec{B}_{\text{wire}} + \vec{B}_{\text{E}}$. The deflection angle is set by the ratio of magnitudes: $\tan\theta = B_{\text{wire}}/B_{\text{E}} = 80/20 = 4$, hence $\theta \approx 76^\circ$. This follows directly from the fields of two independent sources adding as vectors.

A lightning channel carries a current $I = 3\cdot10^{4}$ A. At a point $r = 100$ m from the channel the induction is $B = \frac{\mu_0 I}{2\pi r} = \frac{2\cdot10^{-7}\cdot3\cdot10^{4}}{100} = 6\cdot10^{-5}$ T $= 60$ µT. The Earth's field nearby is $B_{\text{E}} \approx 5\cdot10^{-5}$ T $= 50$ µT. The main stroke lasts about $10^{-4}$ s.

What matters here is the comparison of scales: 60 µT and 50 µT are fields of the same order, so for opposite directions the sum falls to $|60-50| = 10$ µT, while for the same direction it grows to $60+50 = 110$ µT, and the local field changes several-fold for a moment. This is superposition: fields of independent sources add as vectors instead of replacing one another.