1. For an electromagnetic wave in free space, the instantaneous energy densities are:

$u_E = \frac{1}{2}\varepsilon_0 E^2$

and

$u_B = \frac{1}{2\mu_0} B^2$

2. In a free-space electromagnetic wave, the electric and magnetic fields are related by:

$E = cB$

where

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

3. Substitute $B = \frac{E}{c}$ into the magnetic energy density:

$u_B = \frac{1}{2\mu_0} \left(\frac{E}{c}\right)^2$

Using

$c^2 = \frac{1}{\mu_0 \varepsilon_0}$

we get

$u_B = \frac{1}{2\mu_0} E^2 (\mu_0 \varepsilon_0) = \frac{1}{2}\varepsilon_0 E^2$

Thus,

$u_B = u_E$

4. Since the instantaneous electric and magnetic energy densities are equal at every instant, their average values are also equal:

$U_e = U_m$

5. Checking options:

• A: $U_e = U_m$ ✅
• B: $U_e \ne U_m$ ❌
• C: $U_e < U_m$ ❌
• D: $U_e > U_m$ ❌

Therefore, the correct answer is A.