Magnetic Energy in Solenoid
(a) Obtain the expression for the magnetic energy stored in a solenoid in terms of magnetic field B, area A and length l of the solenoid.
(b) Compare this magnetic energy with the electrostatic energy stored in a capacitor.
\[U_B = \frac{1}{2\mu_0} B^2 Al\]
Magnetic energy density:
\[u_B = \frac{B^2}{2\mu_0}\]
Electrostatic energy density:
\[u_E = \frac{1}{2} \epsilon_0 E^2\]
The magnetic energy stored in a solenoid is analogous to the electric energy stored in a capacitor. Both represent energy stored in their respective fields.
Interactive Simulation
Adjust the parameters below to explore how magnetic and electric energy storage systems compare:
Energy Storage Comparison
Both solenoids and capacitors store energy in their respective fields. The energy densities show fundamental similarities:
The energy density in a magnetic field depends on the square of the magnetic field strength (B) and the magnetic permeability of free space (μ₀):
For our solenoid with B = 0.05 T:
\[u_B = \frac{(0.05)^2}{2(4π×10^{-7})} = 0.00 \text{ J/m}^3\]
The energy density in an electric field depends on the square of the electric field strength (E) and the electric permittivity of free space (ε₀):
For our capacitor with E = 5.0 × 10⁵ V/m:
\[u_E = \frac{1}{2}(8.854×10^{-12})(5.0 × 10⁵)^2 = 0.00 \text{ J/m}^3\]
This comparison shows that while the forms are similar, the actual energy densities depend on the field strengths and the fundamental constants μ₀ and ε₀, which are related through the speed of light: μ₀ε₀ = 1/c².
