14.3 Energy in a Magnetic Field

Magnetic fields store energy when electric current flows through a conductor. This energy is crucial in various applications, from power lines to electromagnets. Understanding how magnetic fields store and transfer energy is key to grasping their role in electrical systems.

The energy stored in magnetic fields can be quantified using formulas that relate to field strength and volume. This concept extends to inductors, where the energy storage depends on inductance and current. These principles underpin the functioning of transformers and electromagnets in our everyday lives.

Energy in Magnetic Fields

Energy storage in magnetic fields

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• Magnetic fields store energy when an electric current flows through a conductor
  • Energy is stored in the space surrounding the conductor (power lines, electromagnets)
• Magnetic field energy density formula:
  • $u_B = \frac{B^2}{2\mu_0}$
    • $u_B$ represents energy density in J/m³
    • $B$ represents magnetic field strength in teslas (T)
    • $\mu_0$ represents permeability of free space, equal to $4\pi \times 10^{-7}$ T⋅m/A
• Total energy stored in a magnetic field calculated by volume integral of energy density:
  • $U_B = \int u_B dV = \int \frac{B^2}{2\mu_0} dV$
    • $U_B$ represents total energy stored in the magnetic field in joules (J)
    • $dV$ represents differential volume element in m³

Equation for inductor energy

• Inductor is a passive electrical component that stores energy in its magnetic field when electric current flows through it (solenoid, transformer)
• Magnetic field strength inside an inductor is proportional to the current:
  • $B = \mu_0 n I$
    • $n$ represents number of turns per unit length of the inductor in turns/m
    • $I$ represents current flowing through the inductor in amperes (A)
• Substituting magnetic field strength expression into energy density formula:
  • $u_B = \frac{(\mu_0 n I)^2}{2\mu_0} = \frac{\mu_0 n^2 I^2}{2}$
• Total energy stored in an inductor calculated by volume integral of energy density:
  • $U_L = \int \frac{\mu_0 n^2 I^2}{2} dV = \frac{\mu_0 n^2 I^2}{2} \int dV = \frac{\mu_0 n^2 I^2}{2} A l = \frac{L I^2}{2}$
    • $A$ represents cross-sectional area of the inductor in m²
    • $l$ represents length of the inductor in m
    • $L = \mu_0 n^2 A l$ represents inductance of the inductor in henries (H)

Applications of magnetic energy storage

• Electromagnets
  • Consist of a coil of wire wrapped around a ferromagnetic core (iron, steel)
  • Electric current flowing through the coil generates a magnetic field
  • Magnetic field strength depends on the current and number of turns in the coil
  • Stored magnetic energy used to perform work (lifting heavy objects, magnetic levitation trains)
• Transformers
  • Consist of two or more coils of wire wound around a common ferromagnetic core
  • Alternating current in the primary coil creates a changing magnetic field in the core
  • Changing magnetic field induces an alternating voltage in the secondary coil
  • Voltage transformation ratio determined by the ratio of the number of turns in the primary and secondary coils
  • Energy transferred from the primary to the secondary coil through the shared magnetic field in the core
  • Used to step up or step down voltages in power transmission and distribution systems (power grid, electronic devices)

Magnetic properties of materials

• Magnetic flux: The total magnetic field passing through a given area
• Magnetic dipole moment: A measure of the torque experienced by a magnetic dipole in an external magnetic field
• Magnetic susceptibility: A measure of how easily a material can be magnetized in response to an external magnetic field
• Magnetization: The process by which materials form magnetic dipoles when exposed to a magnetic field
• Magnetic permeability: A measure of a material's ability to support the formation of a magnetic field within itself