Inductance, Magnetic Energy and Transients
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You must be comfortable with Faraday’s law of induction, the definition of magnetic flux, and basic first-order differential equations. Familiarity with energy conservation in mechanical systems is also assumed.
Why this matters
Inductance is the electromagnetic analogue of inertia, governing how circuits resist changes in current.
In statics, we treated electric and magnetic fields as separate entities. In dynamics, they are coupled. Inductance quantifies the magnetic field energy stored by a current-carrying circuit. Understanding this storage mechanism is essential for analysing transients, AC circuits, and the propagation of electromagnetic waves. Without the concept of inductance, we cannot explain why a simple switch does not produce an instantaneous current, nor can we derive the wave equation that underpins all modern communication technology.
The Idea of Inductance
Imagine a circuit with a battery and a resistor. When you close the switch, the current does not jump instantly to its maximum value. Instead, it rises gradually. Why? Because as current begins to flow, it generates a magnetic field around the wires. This changing magnetic field creates an electric field that opposes the change in current, a direct consequence of Faraday’s law. This opposition is not a frictional force that dissipates energy as heat; rather, it is a storage mechanism. The energy supplied by the battery is temporarily stored in the magnetic field. Inductance is the proportionality constant that relates the magnetic flux linkage to the current. It is the circuit’s "magnetic inertia."
Definition and Mutual Inductance
For a single loop, the self-inductance \(L\) is defined by the relationship between the total magnetic flux \(\Phi_B\) through the loop and the current \(I\) producing it. If the medium is linear and isotropic, the flux is proportional to the current:
Here, \(L\) depends only on the geometry of the circuit and the permeability of the surrounding medium, not on the current itself. For two coupled circuits, the flux in circuit 1 due to the current in circuit 2 is \(\Phi_{12} = M I_2\), where \(M\) is the mutual inductance. By reciprocity, \(M_{12} = M_{21} = M\). The total flux in circuit 1 is then \(\Phi_1 = L_1 I_1 + M I_2\). The induced electromotive force (emf) in circuit 1 is given by Faraday’s law:
The term \(-L_1 \frac{dI_1}{dt}\) is the self-induced emf, and \(-M \frac{dI_2}{dt}\) is the mutually induced emf. The negative sign reflects Lenz’s law: the induced emf opposes the change in flux that produced it.
Magnetic Energy Storage
When current builds up in an inductor, work is done against the induced emf. The power delivered to the inductor is \(P = I \mathcal{E}_{ind} = I \left( L \frac{dI}{dt} \right)\). The energy \(U_B\) stored in the magnetic field is the integral of this power over time:
This energy is distributed throughout the volume occupied by the magnetic field. For a general magnetic field \(\mathbf{B}\) in a linear medium with permeability \(\mu\), the energy density \(u_B\) is:
Thus, the total magnetic energy is the volume integral of this density:
This expression is fundamental. It shows that the energy is not in the wires, but in the field itself. This field-centric view is crucial for understanding radiation and wave propagation, where energy flows through space via the Poynting vector.
RL Transients
Consider a series circuit with a resistor \(R\), an inductor \(L\), and a DC voltage source \(V_0\). Applying Kirchhoff’s voltage law, we have:
Rearranging gives a first-order linear differential equation:
The solution, assuming \(I(0) = 0\), is:
where \(\tau = L/R\) is the time constant. The current approaches its steady-state value \(V_0/R\) exponentially. The energy stored in the inductor at time \(t\) is \(U_B(t) = \frac{1}{2} L I(t)^2\). Initially, all energy from the source is stored in the magnetic field. As \(t \to \infty\), the inductor behaves like a short circuit, and the energy remains constant in the field while the source continues to supply power to the resistor.
The derivations
Faraday's Law and the Flux Rule
For any circuit C bounding a surface S, the electromotive force equals minus the rate of change of magnetic flux, ℰ = −dΦB/dt, and this single "flux rule" resolves into two physically distinct contributions — the induced curl of the electric field, giving the differential law ∇ × E = −∂B/∂t, and the motional term ∮ (v × B)·dl arising from the Lorentz force on charges carried by a moving or deforming conductor.
AC Circuits: Phasors, Impedance and Series RLC Resonance
For a linear series RLC circuit driven by a sinusoidal voltage source \(V(t)=V_0 \cos(\omega t)\), the steady-state current is \(I(t)=I_0 \cos(\omega t - \phi)\) where the amplitude \(I_0\) and phase lag \(\phi\) are determined by the complex impedance \(Z = R + i\left(\omega L - \frac{1}{\omega C}\right)\). Specifically, \(I_0 = \frac{V_0}{|Z|}\) and \(\tan\phi = \frac{\omega L - 1/(\omega C)}{R}\). The system resonates at \(\omega_0 = \frac{1}{\sqrt{LC}}\), where the impedance is purely resistive (\(Z=R\)) and the current amplitude is maximal.
A common misconception is that inductance is a property of the wire material. In fact, \(L\) is purely geometric and depends on the permeability of the space around the circuit. If you replace an air-core inductor with a ferromagnetic core, \(L\) increases significantly because \(\mu\) increases, but the energy density formula \(u_B = B^2/2\mu\) must be applied carefully if the material is non-linear or hysteretic. For linear media, the energy is always positive, but the sign of mutual inductance \(M\) depends on the relative orientation of the coils (dot convention). Getting the sign wrong leads to incorrect transient behaviour in coupled circuits. C
Worked example
Problem. An air-cored solenoid has 500 turns, length 0.50 m and cross-sectional area \(2.0 \times 10^{-4}\ \mathrm{m^2}\). It is connected through a \(10\ \Omega\) resistor to a 12 V battery. Find \(L\), the time constant, the steady current, the stored energy, and the time for the current to reach 90% of its final value.
- Inductance: \(L = \mu_0 N^2 A/l = 4\pi \times 10^{-7} \times 500^2 \times 2.0 \times 10^{-4}/0.50 = 1.26 \times 10^{-4}\ \mathrm{H}\).
- Time constant: \(\tau = L/R = 1.26 \times 10^{-4}/10 = 1.26 \times 10^{-5}\ \mathrm{s}\).
- Steady current: \(I_\infty = V_0/R = 12/10 = 1.2\ \mathrm{A}\).
- Stored energy: \(U_B = \tfrac12 L I_\infty^2 = 0.5 \times 1.26 \times 10^{-4} \times 1.44 = 9.0 \times 10^{-5}\ \mathrm{J}\).
- 90% rise: \(1 - e^{-t/\tau} = 0.90\) gives \(t = \tau \ln 10 = 2.30 \times 1.26 \times 10^{-5} = 2.9 \times 10^{-5}\ \mathrm{s}\), about 29 μs.
Units check. \(\mathrm{H\,m^{-1}} \times \mathrm{m^2}/\mathrm{m} = \mathrm{H}\); \(\mathrm{H}/\Omega = \mathrm{s}\); \(\mathrm{H\,A^2} = \mathrm{J}\).
Three quick tests
- If the current in an inductor is constant, what is the voltage across it?
Answer
Zero. Since \(V = L \frac{dI}{dt}\), if \(dI/dt = 0\), then \(V = 0\). - Does the energy stored in an inductor depend on the resistance of the circuit?
Answer
No. The energy \(U_B = \frac{1}{2} L I^2\) depends only on inductance and current. Resistance affects how quickly the current reaches its steady value, but not the final stored energy for a given current. - Why is the energy density of the magnetic field proportional to \(B^2\)?
Answer
It arises from the work done to establish the field. The force required to increase the current is proportional to the existing flux (and thus \(B\)), and the work is the integral of force over displacement (current), leading to a quadratic dependence.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-160 · D-429 · Threads: force, energy, light