Problem — Energy in the field of a solenoid

Problem

Estimate the energy stored in the magnetic field of a large laboratory solenoid with inductance $L=0{,}1\ \text{H}$ carrying a current $i=3\ \text{A}$. Compare the result with the kinetic energy of an apple ($m\approx 0{,}1\ \text{kg}$) thrown at $v=3\ \text{m/s}$.

Solution

Magnetic energy of the solenoid. The energy stored by an inductance is: $E_L=\frac{1}{2}L\,i^2=\frac{1}{2}\cdot 0{,}1\cdot 3^2=\frac{1}{2}\cdot 0{,}1\cdot 9=0{,}45\ \text{J}\approx 0{,}5\ \text{J}$

Kinetic energy of the apple. $E_\text{kin}=\frac{1}{2}m\,v^2=\frac{1}{2}\cdot 0{,}1\cdot 3^2=0{,}45\ \text{J}$

The two energies are comparable, both of the order of $0{,}5\ \text{J}$. A “large” laboratory solenoid thus stores, in its magnetic field, an energy comparable to the kinetic energy of an apple thrown by hand: a modest but not negligible amount.

$\ev{E_L=\tfrac{1}{2}L\,i^2\approx 0{,}5\ \text{J}\approx E_\text{kin}\ \text{of the apple}}$

Links

Topics: Electromagnetic induction Concepts: Inductance and self-induction · Field energy density Objects: Solenoid