During the propagation of electromagnetic waves in a medium,
(a) electric energy density is double o

Q: Why do we use $\varepsilon$ and $\mu$ instead of $\varepsilon_0$ and $\mu_0$?

Because the question specifically mentions propagation in a *medium*. $\varepsilon_0$ and $\mu_0$ are strictly for a vacuum. For any other medium, we must use its specific permittivity $\varepsilon$ and permeability $\mu$.

Q: Does $u_E = u_B$ mean the electric and magnetic fields have the same magnitude?

No! The magnitudes $E$ and $B$ are very different and even have different units. The equality $u_E = u_B$ only means that the *energy* stored in each field per unit volume is the same.

Q: What is the total energy density of the electromagnetic wave?

The total energy density $u$ is the sum of the electric and magnetic energy densities. Since they are equal, $u = u_E + u_B = 2u_E = 2u_B = \varepsilon E^2 = \frac{B^2}{\mu}$.

Q: Is this equipartition of energy true for standing waves as well?

In a standing wave, the electric and magnetic fields are out of phase by $90^\circ$ in time. So, locally at a specific point and time, $u_E$ may not equal $u_B$. However, their *time-averaged* energy densities over a full cycle are still equal.

Q: How does this energy actually propagate forward?

The propagation of energy is described by the Poynting vector, $\vec{S} = \frac{1}{\mu} (\vec{E} \times \vec{B})$. It points in the direction of wave propagation and its magnitude represents the rate of energy transfer per unit area.

Question

During the propagation of electromagnetic waves in a medium,

Options:

(A) electric energy density is double of the magnetic energy density

(B) electric energy density is half of the magnetic energy density

(C) electric energy density is equal to the magnetic energy density

(D) Both electric and magnetic energy densities are zero

Sigma Percentile · Accepted Answer

### The Anatomy of an Electromagnetic Wave

Imagine a beam of light traveling through a glass slab or a radio wave broadcasting through the atmosphere. These are electromagnetic waves, and as they propagate through any medium, they carry energy. But how is this energy distributed? Is it mostly in the electric field, or the magnetic field?

Let's break down the energy densities. The energy density (energy per unit volume) of an electric field $E$ in a medium with permittivity $\varepsilon$ is given by:
$$u_E = \frac{1}{2}\varepsilon E^2$$

Similarly, the energy density of a magnetic field $B$ in a medium with permeability $\mu$ is:
$$u_B = \frac{B^2}{2\mu}$$

### The Intimate Link Between E and B

In an electromagnetic wave, the electric and magnetic fields are not independent; they are intimately coupled. The magnitude of the electric field is related to the magnetic field by the speed of the wave $v$ in that medium:
$$E = vB$$

Furthermore, the speed of the wave itself is dictated by the electromagnetic properties of the medium:
$$v = \frac{1}{\sqrt{\mu\varepsilon}}$$

### The Beautiful Symmetry: Equipartition of Energy

Let's see what happens when we substitute these relationships into our electric energy density formula. We start with:
$$u_E = \frac{1}{2}\varepsilon E^2$$

Substitute $E = vB$:
$$u_E = \frac{1}{2}\varepsilon (vB)^2 = \frac{1}{2}\varepsilon v^2 B^2$$

Now, substitute $v^2 = \frac{1}{\mu\varepsilon}$:
$$u_E = \frac{1}{2}\varepsilon \left(\frac{1}{\mu\varepsilon}\right) B^2$$

Notice how the permittivity $\varepsilon$ beautifully cancels out! We are left with:
$$u_E = \frac{B^2}{2\mu}$$

But wait, this is exactly the expression for the magnetic energy density $u_B$! 
$$u_E = u_B$$

**This is a profound result:** During the propagation of an electromagnetic wave in any medium, the energy is always perfectly and equally divided between the electric and magnetic fields. This principle is known as the equipartition of energy in electromagnetic waves. Therefore, the electric energy density is equal to the magnetic energy density.

Animated Solution for Physics - Electromagnetic Waves: During the propagation of electromagnetic waves in a medium,

Select Answer:

Visualized Solution

Visualizing the Electromagnetic Wave

Energy Densities

Wave Speed and Field Relations

Equating the Densities

Conclusion

The Way Forward

The Sigma Insight: Characteristics of Electromagnetic Waves

The Anatomy of an Electromagnetic Wave

The Intimate Link Between E and B

The Beautiful Symmetry

Similar Questions

For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric ( $U_{e}$ ) and magnetic ( $U_{m}$ ) fields is

In free space, the energy of electromagnetic wave in electric field is $U_{E}$ and in magnetic field is $U_{B}$ . Then

An electromagnetic wave of intensity $50 Wm^{- 2}$ enters in a medium of refractive index ' $n$ ' without any loss. The ratio of the magnitudes of electric fields and the ratio of the magnitudes of magnetic fields of the wave before and after entering into the medium are respectively, given by

Electromagnetic waves are transverse in nature is evident by

An electromagnetic wave of frequency $ν = 3.0 MHz$ passes from vacuum into a dielectric medium with permittivity $ε = 4.0$ . Then,

An electromagnetic wave in vacuum has the electric and magnetic fields $E$ and $B$ , which are always perpendicular to each other. The direction of polarisation is given by $X$ and that of wave propagation by $k$ . Then,

The electric field of a plane electromagnetic wave is given by $E = E_{0} (\hat{x} + \hat{y}) sin (k z - ω t)$ Its magnetic field will be given by

A plane electromagnetic wave propagating along $y$ -direction can have the following pair of electric field ( $E$ ) and magnetic field ( $B$ ) components.

The electric field of a plane electromagnetic wave propagating along the x-direction in vacuum is $E = E_{0} \hat{j} cos (ω t - k x)$ . The magnetic field $B$ , at the moment $t = 0$ is