Question

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

Answer 1

The ratio of the energy density of the magnetic field to that of the electric field inside a current-carrying wire is given by the expression µ₀/ε₀ (I L /R 2πr)², which corresponds to option (a).

Step-by-step explanation:

To calculate the ratio of the energy density of the magnetic field to that of the electric field just inside a wire carrying current I, we need to use the energy density formulas for both fields. The energy density of the magnetic field (um) is given by um = B2/(2μ0), where B is the magnetic field strength, and μ0 is the permeability of free space. The energy density of the electric field (ue) is given by ue = ε0E2/2, where E is the electric field strength, and ε0 is the permittivity of free space.

The magnetic field strength B inside a long wire due to a current I can be determined using Ampere's Law. For a wire of radius r and length L carrying current I, the field strength at a distance r from the center is given by B = (μ0I)/(2πr). Considering this, the ratio of the magnetic to electric energy densities (um/ue) can be calculated and simplified to option (a):

μ0/ε0 (I L /R 2πr)2

Answer 2

`(d) \ \ (\mu_o)/(\epsilon_o) ((L)/(2\pi r*R) )^2`

Step-by-step explanation:

Energy density in magnetic field is given as;

`U_B = (1)/(2 \mu_o) B^2`

where;

B is the magnetic field strength

Energy density of electric field

`U_E = (1)/(2)\epsilon E^2`

where;

E is electric field strength

Take the ratio of the two fields energy density

`(U_B)/(U_E) = (1)/(2\mu_o) B^2 / (1)/(2)\epsilon E^2\\\\(U_B)/(U_E) = (B^2)/(2\mu_o) *(2)/(\epsilon E^2) \\\\(U_B)/(U_E) = (1)/(\mu_o \epsilon) ((B^2)/(E^2))`

`(U_B)/(U_E) = (1)/(\mu_o \epsilon) ((B)/(E))^2`

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

`(U_B)/(U_E) = (1)/(\mu_o \epsilon) ((B*L)/(E*L))^2`

Now replace E x L with IR

`(U_B)/(U_E) = (1)/(\mu_o \epsilon) ((B*L)/(IR))^2`

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

`(U_B)/(U_E) = (1)/(\mu_o \epsilon) ((\mu_oI*L)/(2\pi r* IR))^2`

cancel out the current "I" and factor out μ₀

`(U_B)/(U_E) = (\mu_o^2)/(\mu_o \epsilon) ((L)/(2\pi r* R))^2`

Finally, the equation becomes;

`(U_B)/(U_E) = (\mu_o)/(\epsilon) ((L)/(2\pi r*R ))^2`

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