Mutual Induction & the Induction Coil
One circuit's changing current drives another.
Lesson at a glanceShow the key points
- Φ₂ = Mi₁ (38.14); ℰ₂ = −M di₁/dt (38.15); M is symmetric.
- Coaxial solenoids: M = μ₀n₁n₂πr₁²l.
- Charge through a coupled coil depends only on the total flux change.
- Induction coil: slow make, fast break; capacitor speeds the break and reverses the current.
A changing current in one circuit can induce an emf in a completely separate circuit nearby. That coupling is the heart of transformers, induction coils and wireless chargers.
Mutual inductance
Current i in one circuit produces a field whose flux through a second circuit is proportional to i:
For circuits with several turns, add the flux through every turn before writing Φ = Mi.
A solenoid of radius 1.5 cm and 15 turns/cm sits inside a longer solenoid with 8 turns/cm. Find their mutual inductance over a 40 cm length.
A long solenoid of radius 1.5 cm with 80 turns/cm is surrounded by a 50-turn coil of total resistance 10 Ω connected to a galvanometer. The solenoid current is reversed from 4.0 A one way to 4.0 A the other. How much charge passes through the galvanometer?
The induction coil
Ruhmkorff’s induction coil makes a very high emf from a low-voltage cell. A primary coil T is wound on a laminated soft-iron core; a secondary coil S of many turns is wound over it and ends in two rods G₁ and G₂ with an adjustable gap. The primary circuit has a cell and an automatic interrupter: a springy metal strip A carrying a soft-iron hammer H, touching a screw D. A capacitor C is connected across the interrupter.
How it works
- Make and break. Current through the primary magnetizes the core, which pulls the hammer and breaks the contact. The current stops, the core lets go, the strip springs back and the contact closes again, over and over.
- Slow rise, fast fall. On make, the primary’s self-inductance makes the current grow slowly. On break, it collapses very quickly.
- Secondary emf. The secondary emf −M di/dt is therefore small on make and very large on break (opposite in sign), enough to throw sparks across the gap: about 50 000 V from a 12 V cell, enough to run a discharge tube.
- The capacitor. On break, the primary’s own induced emf would spark across the interrupter, keeping current flowing, slowing its fall and pitting the contacts. The capacitor absorbs that surge instead, so the current drops faster; it then discharges back through the primary, swinging the current almost to −i. The change in flux nearly doubles and so does the secondary emf.
- 1Φ₂ = Mi₁ (38.14); ℰ₂ = −M di₁/dt (38.15); M is symmetric between the two circuits.
- 2Coaxial solenoids: M = μ₀n₁n₂πr₁²l.
- 3Induction coil: slow make, fast break; the break gives a huge secondary emf; the capacitor speeds the break and reverses the current.
Check yourself
Keep exploring
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