Which of the following expressions gives the ratio of the energy density of the magnetic field to that of the electric field just inside the surface of a wire of length L, radius r, and resistance R which is carrying a current I?

a) μ₀/ϵ₀ (I L /R 2πr)²
b) 1/μ₀ ϵ₀ (I L /R 2πr)²
c) 1/μ₀ ϵ₀ (L /R 2πr)²
d) μ₀/ϵ₀ (L /R 2πr)²

Explanation:

Energy density in magnetic field is given as;

where;

B is the magnetic field strength

Energy density of electric field

where;

E is electric field strength

Take the ratio of the two fields energy density

But, Electric field potential, V = E x L = IR (I is current and R is resistance)

Now replace E x L with IR

Also, B = μ₀I / 2πr, substitute this value in the above equation

cancel out the current "I" and factor out μ₀

Finally, the equation becomes;

Therefore, the correct option is (d) μ₀/ϵ₀ (L /R 2πr)²