Answer:
![(d) \ \ \frac{\mu_o}{\epsilon_o} (\frac{L}{2\pi r*R} )^2](https://tex.z-dn.net/?f=%28d%29%20%5C%20%5C%20%5Cfrac%7B%5Cmu_o%7D%7B%5Cepsilon_o%7D%20%28%5Cfrac%7BL%7D%7B2%5Cpi%20r%2AR%7D%20%29%5E2)
Explanation:
Energy density in magnetic field is given as;
![U_B = \frac{1}{2 \mu_o} B^2](https://tex.z-dn.net/?f=U_B%20%3D%20%5Cfrac%7B1%7D%7B2%20%5Cmu_o%7D%20B%5E2)
where;
B is the magnetic field strength
Energy density of electric field
![U_E = \frac{1}{2}\epsilon E^2](https://tex.z-dn.net/?f=U_E%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cepsilon%20E%5E2)
where;
E is electric field strength
Take the ratio of the two fields energy density
![\frac{U_B}{U_E} = \frac{1}{2\mu_o} B^2 / \frac{1}{2}\epsilon E^2\\\\\frac{U_B}{U_E} = \frac{B^2}{2\mu_o} *\frac{2}{\epsilon E^2} \\\\\frac{U_B}{U_E} = \frac{1}{\mu_o \epsilon} (\frac{B^2}{E^2})](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B2%5Cmu_o%7D%20B%5E2%20%2F%20%5Cfrac%7B1%7D%7B2%7D%5Cepsilon%20E%5E2%5C%5C%5C%5C%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7BB%5E2%7D%7B2%5Cmu_o%7D%20%20%2A%5Cfrac%7B2%7D%7B%5Cepsilon%20E%5E2%7D%20%5C%5C%5C%5C%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7BB%5E2%7D%7BE%5E2%7D%29)
![\frac{U_B}{U_E} = \frac{1}{\mu_o \epsilon} (\frac{B}{E})^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7BB%7D%7BE%7D%29%5E2)
But, Electric field potential, V = E x L = IR (I is current and R is resistance)
![\frac{U_B}{U_E} = \frac{1}{\mu_o \epsilon} (\frac{B*L}{E*L})^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7BB%2AL%7D%7BE%2AL%7D%29%5E2)
Now replace E x L with IR
![\frac{U_B}{U_E} = \frac{1}{\mu_o \epsilon} (\frac{B*L}{IR})^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7BB%2AL%7D%7BIR%7D%29%5E2)
Also, B = μ₀I / 2πr, substitute this value in the above equation
![\frac{U_B}{U_E} = \frac{1}{\mu_o \epsilon} (\frac{\mu_oI*L}{2\pi r* IR})^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7B%5Cmu_oI%2AL%7D%7B2%5Cpi%20r%2A%20IR%7D%29%5E2)
cancel out the current "I" and factor out μ₀
![\frac{U_B}{U_E} = \frac{\mu_o^2}{\mu_o \epsilon} (\frac{L}{2\pi r* R})^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B%5Cmu_o%5E2%7D%7B%5Cmu_o%20%5Cepsilon%7D%20%28%5Cfrac%7BL%7D%7B2%5Cpi%20r%2A%20R%7D%29%5E2)
Finally, the equation becomes;
![\frac{U_B}{U_E} = \frac{\mu_o}{\epsilon} (\frac{L}{2\pi r*R })^2](https://tex.z-dn.net/?f=%5Cfrac%7BU_B%7D%7BU_E%7D%20%3D%20%5Cfrac%7B%5Cmu_o%7D%7B%5Cepsilon%7D%20%28%5Cfrac%7BL%7D%7B2%5Cpi%20r%2AR%20%7D%29%5E2)
Therefore, the correct option is (d) μ₀/ϵ₀ (L /R 2πr)²