If the magnetic energy density u_B is doubled due to a change in magnetic field, by what factor does u_B increase when B doubles?

If the magnetic energy density u_B is doubled due to a change in magnetic field, by what factor does u_B increase when B doubles?

• A. It increases by a factor of 4
• B. It increases by a factor of 2
• C. It remains the same
• D. It decreases by a factor of 2

Answer: (A)

The magnetic energy density grows with the square of the magnetic field. In SI units, u_B = B^2/(2μ0). So when the magnetic field doubles (B → 2B), the energy density becomes (2B)^2/(2μ0) = 4B^2/(2μ0) = 4u_B. Doubling B thus increases u_B by a factor of four. This shows why the energy stored in a magnetic field is very sensitive to changes in the field strength.

The magnetic energy density grows with the square of the magnetic field. In SI units, u_B = B^2/(2μ0). So when the magnetic field doubles (B → 2B), the energy density becomes (2B)^2/(2μ0) = 4B^2/(2μ0) = 4u_B. Doubling B thus increases u_B by a factor of four. This shows why the energy stored in a magnetic field is very sensitive to changes in the field strength.