The flux linkage per unit length will be the magnetic flux through the area bounded by the two dotted lines and the wires. Since the wires are symmetric about this area, you may calculate the flux linked from just one wire and then double. The formula found for the magnetic field about a long wire should be used here.
The area of linkage can be spanned with the infinitesimal element shown, varying r from a to d-a. In using these limits of integration, we are staying outside the wires; the magnetic flux linked inside the wire may also be included, but be sure to use the correct expression for the magnetic field inside the sire.
b) Use the same current in i2R.
c) Use the same current in i emf.
a) Integrate the power provided by the battery over the time specified, using the time-dependent current for an energizing inductor.
b) Use U = Li2/2.
c) Find the difference between the answers for a) and b).
Equate the two energy densities and solve for E. Notice the role of the permittivity and permeability in the final expression.
a) Use the value for the magnetic field in a toroid in the expression for magnetic energy density.
b) The energy density should be integrated using circular strips of radius r, height h, and thickness dr, which has an infinitesimal volume of 2 pi r h dr.