Concept understanding — Self-Inductance of a Solenoid
Self-Inductance of a Solenoid: From Intuition to Formula
Imagine you push a heavy door. It doesn't resist your push once it's moving — but it does resist you trying to change its speed suddenly. That resistance to change is inertia. A solenoid carrying current behaves the same way: it "wants" to keep its current steady, and fights any attempt to change it.
This property is called self-inductance. The solenoid generates a back emf that opposes the change in its own current — not the current itself, but the change in current. That's the core idea.
Why does a solenoid oppose current changes?
A solenoid is a long coil of wire. When current flows through it, it produces a magnetic field inside. If you try to increase the current, the magnetic field strengthens. But a changing magnetic field induces an emf in the coil itself (Faraday's law). By Lenz's law, this induced emf opposes the change that caused it — so it pushes back against the rising current.
If you try to decrease the current, the field weakens, and the induced emf tries to keep the current flowing. The solenoid acts like an electrical "flywheel."
The precise statement
Self-inductance L is defined by the relation:
E=−LdtdI
where E is the induced back emf, and dtdI is the rate of change of current. The negative sign tells you the emf opposes the change.
For a solenoid, L depends only on its geometry and the core material — not on the current. The formula is:
L=μ0n2Al
L=μ0n2Al
Let's unpack each symbol:
μ0 — permeability of free space (4π×10−7 H/m). It's a universal constant that tells you how strongly a vacuum responds to magnetic fields.
n — number of turns per unit length (turns/m). More turns per metre means a stronger field per ampere, so more inductance.
A — cross-sectional area of the solenoid (m²). A wider coil encloses more magnetic flux.
l — length of the solenoid (m). Longer solenoid means more total turns, hence more inductance.
Where does L=μ0n2Al come from?
Start with the magnetic field inside a long solenoid:
B=μ0nI
The magnetic flux through one turn is BA=μ0nIA. For all N=nl turns, the total flux linkage is:
Φtotal=N⋅BA=(nl)(μ0nIA)=μ0n2AlI
By definition, self-inductance is the constant of proportionality between flux linkage and current:
Φtotal=LI
Comparing, you get:
L=μ0n2Al
Note
This formula assumes an ideal solenoid — infinitely long, with a uniform field inside and zero field outside. Real solenoids are close approximations if l≫A.
What does a larger L mean?
A solenoid with high L strongly resists changes in current. If you try to switch the current on quickly, the back emf is large, so the current rises slowly. If you short-circuit the solenoid, the current doesn't drop instantly — it decays gradually.
This is why inductors are used in filters, chokes, and timing circuits. They smooth out current variations. …
The self-inductance is found using Faraday’s law for a changing current: L=∣ΔI/Δt∣∣E∣. With E=200V, ΔI=−5.0A, and Δt=0.1s, we get L=4.0H.
The key idea here is self-inductance — a circuit’s property that opposes a change in current by inducing an emf. When the current changes, the magnetic flux through the circuit itself changes, and that induces an emf (back emf) given by:
E=−LdtdI
The negative sign is Lenz’s law: the induced emf opposes the change. But for magnitude, we drop the sign and use the average values.
Since the current falls uniformly from 5.0A to 0.0A in 0.1s, the average rate of change is:
ΔtΔI=0.10.0−5.0=0.1−5.0=−50A/s
The magnitude of this rate is 50A/s.
The average induced emf is given as 200V. Using the magnitude form of Faraday’s law:
Mistake 1: Forgetting the Negative Sign in Faraday's Law
The mistake: Students often write:
ε=LΔtΔI
and plug in values without the sign, getting confused about the answer.
Why it's wrong: The correct relation is:
ε=−LdtdI
The negative sign indicates Lenz's law — the induced emf opposes the change in current. When current decreases (dtdI is negative), the induced emf is positive (it tries to keep current flowing).
How to avoid: Always write the full equation with the sign. Then, when using magnitudes, take absolute values:
∣ε∣=LΔtΔI
Mistake 2: Using ΔI=5.0A Instead of the Change
The mistake: Some students take ΔI=5.0A (the final value) or get confused about the direction of change.
Why it's wrong: The change in current is:
ΔI=Ifinal−Iinitial=0.0−5.0=−5.0A
The magnitude of change is ∣ΔI∣=5.0A.
How to avoid: Always compute ΔI=If−Ii explicitly. For magnitude problems, use ∣ΔI∣.
Mistake 3: Confusing Δt with Time Constant or Period
The mistake: Students think 0.1s is the time constant (τ=L/R) or the period of oscillation.
Why it's wrong: Here, 0.1s is simply the time interval over which the current changes. It has nothing to do with circuit time constants.
How to avoid: Read the problem carefully. The phrase "falls from ... to ... in 0.1 s" clearly indicates a time interval Δt, not a time constant.
Mistake 4: Incorrect Unit Handling
The mistake: Mixing up units — writing L=5/0.1200 without tracking units.
Why it's wrong: This leads to errors in the final unit (should be henry, not ohm or volt-second).