31.5
Energy In A Magnetic Field
If a magnetic field is sustained, there must be a current in a closed circuit or loop, implying some energy has been spent in creating the field. If t…
Consider an inductor connected to a variable source of EMF. The inductor opposes any change in the current passing through it; hence, an EMF is induced across it.
Now, consider an ideal coil with no resistance; no energy is dissipated across it. Hence, some amount of the energy supplied by the source is stored in the inductor.
The instantaneous power is given by the product of the instantaneous induced EMF and the instantaneous current. The energy supplied to the inductor is its product with the differential time. Assuming that the energy stored is zero when there is no current, integrating the expression gives the energy stored in the magnetic field.
In an ideal toroidal solenoid, assuming a small cross-sectional area through which the magnetic field is constant, the inductance is known and so is its volume. Thus, the magnetic energy density can be calculated.
If the material inside the toroid is not a vacuum but given by the magnetic permeability μ, the expression is modified thus.
View the full transcript and gain access to JoVE Core videos
Q1: How is energy stored in an inductor's magnetic field?
Energy is stored in an inductor when current flows through it. The instantaneous power supplied equals the product of induced EMF and instantaneous current. Integrating this power over time yields the total energy stored in the magnetic field. In an ideal inductor with zero resistance, no energy is dissipated, so all supplied energy is stored in the field.
Q2: What is magnetic energy density and how is it calculated?
Magnetic energy density is the energy per unit volume stored in a magnetic field. For a toroidal solenoid with constant magnetic field across its cross-section, energy density is calculated by dividing total stored energy by the coil's volume. If the material inside has magnetic permeability μ different from vacuum, this permeability replaces the vacuum permeability in the expression.
Q3: Why does an inductor oppose changes in current?
An inductor opposes current changes by generating an induced EMF across itself. This induced EMF resists the change in current flow, following Lenz's law. The opposition becomes more pronounced with rapid current changes, making inductors useful for controlling current dynamics in circuits experiencing current growth and decay.
Q4: How does an automobile ignition system use magnetic field energy?
The ignition system stores energy in primary coils connected to the car's battery, creating a strong magnetic field. When the current is interrupted, the magnetic field collapses rapidly, inducing a high voltage of tens of thousands of volts in secondary coils. This voltage pulse ignites the fuel-air mixture through spark plugs.
Q5: What is the relationship between instantaneous power and energy in an inductor?
Instantaneous power in an inductor is the product of induced EMF and instantaneous current at any moment. Integrating this instantaneous power over time gives the total energy stored. This relationship shows how power flow over time accumulates as stored magnetic field energy in the inductor.
Q6: How does material permeability affect magnetic energy storage?
The magnetic permeability of material inside an inductor directly affects energy storage capacity. In a vacuum, the expression uses the vacuum permeability constant. When a material with different magnetic permeability replaces the vacuum, that material's permeability substitutes into the energy density expression, modifying the total energy stored.
Q7: Why is an ideal inductor with zero resistance important for understanding magnetic energy?
An ideal inductor with zero resistance allows all supplied energy to be stored in the magnetic field without dissipation as heat. This simplification enables clear analysis of energy storage mechanisms. Although practically impossible, assuming negligible resistance makes the mathematical relationships between power, current, and stored energy transparent and measurable.