Self-Inductance and Energy Stored in a Magnetic Field
A coil opposes changes in its own current through its own flux linkage. The energy stored through this opposition can be described as an energy density throughout the field. Inductance is the magnetic partner of capacitance.
What a learner can do afterwards
- Calculates the self-inductance of a solenoid from its geometry
- Finds the energy stored in an inductor and writes it as an energy density in the field
- Describes the current growth in an RL circuit and identifies its time constant
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A coil's current produces its own flux linkage. For a fixed linear inductor, linkage = L I and induced emf = -L dI/dt. The minus sign opposes current change. L is measured in henries and depends on geometry and magnetic material. Treating L as constant requires a linear response; some magnetic cores have current-dependent inductance.
For a long air-core solenoid, ignore end effects. With N turns, length l, area A and n = N/l, B = mu0 n I. Each turn links flux B A, so L = N B A / I = mu0 N squared A / l = mu0 n squared A l. At fixed dimensions and material, more turns give larger L.
For constant L, power absorbed by an inductor is L I dI/dt. Integrating means summing small energy gains L i di as current rises from 0 to I. The area under L i plotted against i is a triangle with base I and height L I, giving U = (1/2) L I squared. At fixed L, doubling current quadruples U. This energy is stored in the magnetic field and can later return to the circuit.
In vacuum, or approximately in an air core, magnetic energy density is u = B squared / (2 mu0). For a uniform field in fixed volume V, U = u V. Doubling B throughout that same volume quadruples the energy. This vacuum formula should not be applied unchanged to arbitrary magnetic materials.
Connect a constant DC emf E to series resistance R and an ideal inductor L, initially at zero current. The loop rule balances the source against voltage drops: E = L dI/dt + R I. Its solution is I = (E/R)(1 - exp(-t/tau)), with tau = L/R; exp(z) means e to the power z. As I rises, E - R I and the slope fall. Current starts at zero and approaches E/R. Remove the source but keep a closed path through R: current is I0 exp(-t/tau), where I0 is the current at switching.
At t = tau, the rise from zero reaches 1 - exp(-1), about 0.632 of E/R. That is about 63%, not the final value. The ideal exponential approaches its final value without reaching it at a finite time. After several time constants the remaining difference can be small enough to neglect for a stated tolerance.
For constant L, induced emf is -L dI/dt and stored energy is half L I squared. An ideal RL rise has time constant L/R and reaches about 63% after one tau.
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