11.2 Energy stored in magnetic fields

Magnetic fields store energy, just like electric fields. This energy is crucial for understanding inductors and electromagnetic devices. The amount of energy depends on the field strength and the volume it occupies.

Inductors are key components that store energy in their magnetic fields. The energy stored is proportional to the inductance and the square of the current. This concept is vital for analyzing circuits and electromagnetic systems.

Magnetic Field Energy

Magnetic Energy Density and Field Energy

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• Magnetic energy density $u_B$ represents the energy per unit volume stored in a magnetic field
  • Defined as $u_B = \frac{B^2}{2\mu_0}$, where $B$ is the magnetic field strength and $\mu_0$ is the permeability of free space
  • Measured in units of joules per cubic meter (J/m³)
  • Increases quadratically with the magnetic field strength
• Magnetic field energy $U_B$ is the total energy stored in a magnetic field within a given volume
  • Calculated by integrating the magnetic energy density over the volume: $U_B = \int u_B dV = \int \frac{B^2}{2\mu_0} dV$
  • Depends on the magnetic field strength and the volume occupied by the field
  • Stored energy is proportional to the square of the magnetic field strength

Magnetization Energy

• Magnetization energy is the energy associated with the alignment of magnetic dipoles in a material
  • Occurs when a material is exposed to an external magnetic field
  • Magnetic dipoles tend to align with the applied field, lowering their potential energy
  • The energy required to magnetize a material is stored as magnetization energy
• The magnetization energy density $u_M$ depends on the magnetization $M$ and the applied magnetic field $H$
  • Defined as $u_M = -\mu_0 \int M \cdot dH$
  • The negative sign indicates that the magnetization energy is released when the magnetic dipoles align with the field
• The total magnetization energy $U_M$ is obtained by integrating the magnetization energy density over the volume of the material
  • $U_M = \int u_M dV = -\mu_0 \int M \cdot H dV$

Energy Storage in Inductors

Energy Stored in an Inductor

• An inductor is a passive electronic component that stores energy in its magnetic field when an electric current flows through it
  • The energy is stored in the magnetic field generated by the current
  • The amount of stored energy depends on the inductance $L$ and the current $I$
• The energy stored in an inductor $U_L$ is given by the formula $U_L = \frac{1}{2}LI^2$
  • $L$ is the inductance measured in henries (H)
  • $I$ is the current flowing through the inductor in amperes (A)
  • The stored energy is proportional to the square of the current
• Example: A 100 mH inductor with a current of 2 A stores an energy of $U_L = \frac{1}{2} \times 0.1 \times 2^2 = 0.2$ J

Joule's Law for Magnetic Fields

• Joule's law for magnetic fields relates the energy stored in an inductor to the work done by the magnetic field
  • States that the work done by a magnetic field in establishing a current $I$ in an inductor with inductance $L$ is equal to the energy stored in the inductor
• The work done by the magnetic field $W_m$ is given by $W_m = \int_0^I L I dI = \frac{1}{2}LI^2$
  • The work done is equal to the area under the curve of the magnetic flux linkage $\lambda = LI$ versus the current $I$
• Joule's law for magnetic fields is analogous to Joule's law for electric fields, which relates the work done by an electric field to the energy stored in a capacitor

Magnetic Work

• Magnetic work is the work done by a magnetic field in moving a magnetic dipole or changing the magnetic flux through a loop
  • Can be positive or negative depending on the relative orientation of the magnetic dipole or current loop with respect to the magnetic field
• The magnetic work done on a magnetic dipole $\vec{m}$ in a magnetic field $\vec{B}$ is given by $W_m = -\int \vec{m} \cdot d\vec{B}$
  • The negative sign indicates that work is done by the field on the dipole when they are aligned
• The magnetic work done on a current loop with current $I$ in a changing magnetic field is given by $W_m = -I \int d\Phi_B$, where $\Phi_B$ is the magnetic flux through the loop
  • The work done is equal to the negative of the change in magnetic flux multiplied by the current
• Example: Moving a magnetic dipole from a region of low magnetic field to a region of high magnetic field requires positive work to be done on the dipole, while the reverse process releases energy