Self-Inductance and Energy Stored in a Magnetic Field
A coil opposes changes in its own current through the flux it links with itself, and the energy that opposition stores can be written as an energy density spread through 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
1 · Read
Every coil fights changes in its own current, and self-inductance measures how hard. When current shifts, the coil's own flux shifts with it and induces a back emf equal to L times dI/dt, always opposing the change. The constant L depends only on geometry: for a solenoid it is mu0 times n squared times A times l, so more turns and fatter cores give bigger L. Think of L as electrical inertia, like a flywheel that shrugs off quick changes.
Building current in a coil takes work, and that work parks in the magnetic field. The stored amount is U equals half L I squared, the magnetic twin of the capacitor's half C V squared. Rewritten through the solenoid field, it becomes an energy density of B squared over 2 mu0, a property of the field itself. Draining the coil hands that energy back, which is why opening a live inductive switch throws a spark.
Close a switch on a series resistor plus inductor and the current does not jump. It creeps up as (E over R) times (1 minus e to the minus t over tau), with tau equal to L over R. Early on the coil drops nearly the whole battery voltage as back emf, and only as the rise slows does the resistor take over. Open the switch and the same tau governs the decay. Big L or small R means a sluggish circuit with a long memory.
Attack any RL problem in this order. Spot tau equals L over R first. Read off the final current E over R. After one tau the current has covered about two thirds of its rise, and after a few tau it is essentially done. Never guess the starting current: an inductor holds its current across a switching instant.
A coil opposes current change with back emf L dI/dt, stores half L I squared in its field, and relaxes with time constant L over R.
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