Energy density of electromagnetic wave, class 12

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Have you ever wondered how your radio, TV, or cell phone receives signals and converts them into the music, images, or text that you enjoy? All of these communication technologies are made possible by the fascinating phenomenon of electromagnetic waves.

These waves are made up of oscillating electric and magnetic fields that travel through space at the speed of light, carrying energy with them. The energy carried by an electromagnetic wave is not distributed uniformly but is concentrated in the fields themselves.

The energy density of an electromagnetic wave is a measure of how much energy is stored in a given volume of space, and it is crucial in understanding how these waves interact with matter and can be harnessed for various applications.

In this article, we will explore the concept of the energy density of electromagnetic waves, how it relates to the strength and rms value of the fields, and their definition, and derivation for the formula of energy density. Stay tuned with this article.

Energy density of electromagnetic wave

Electromagnetic waves are a type of wave that propagates through space and are composed of oscillating electric and magnetic fields. These fields store energy, which is carried along by the wave as it travels through space.

Energy density of electromagnetic wave, class 12
Energy density of electromagnetic wave, class 12

The energy density of an electromagnetic wave is a measure of how much energy is stored in a certain volume of space occupied by the wave.

To give you an idea of how energy density works, think about a battery. A battery stores energy in a compact space, so it has a high energy density. On the other hand, a piece of paper may have some energy stored in its chemical bonds, but it’s spread out over a large area, so it has a low energy density.

Similarly, the energy density of an electromagnetic wave depends on how much energy is stored in the electric and magnetic fields per unit volume of space. The strength of these fields depends on factors like the amplitude of the wave and the frequency of the oscillations.

For example, radio waves have a relatively low energy density because their electric and magnetic fields are relatively weak. In contrast, gamma rays have a very high energy density because they have very strong fields and high frequencies.

Energy density of electromagnetic wave definition

Energy density of electromagnetic wave definition: The energy density of an electromagnetic wave refers to the amount of energy per unit volume carried by the wave. It is a measure of the energy stored in the electric and magnetic fields per unit volume that make up the wave.

Energy density of electromagnetic wave formula

The formula for the energy density (u) of an electromagnetic wave is given as:

$$u=\frac{1}{2} \varepsilon_0 E^2+\frac{1}{2 \mu_0} B^2$$

where ε₀ is the electric constant (also known as vacuum permittivity), μ₀ is the magnetic constant (also known as vacuum permeability), E is the electric field strength, and B is the magnetic field strength.

This formula shows that the energy density of an electromagnetic wave is proportional to the square of the electric and magnetic field strengths, and is also dependent on the electric and magnetic constants of vacuum.

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Derive a formula for the energy density of electromagnetic wave

We know that energy is stored in space wherever electric and magnetic fields are present.
In free space, the energy density of a static electric field E is:

$$
u_E=\frac{1}{2} \varepsilon_0 E^2
$$
Again in free space, the energy density of a static magnetic field is
$$
u_B=\frac{1}{2 \mu_0} B^2
$$
The total energy density of the static electric and magnetic fields will be
$$
u=u_E+u_B=\frac{1}{2} \varepsilon_0 E^2+\frac{1}{2 \mu_0} B^2
$$

Average energy density formula of electromagnetic wave

But in an electromagnetic wave, both $E$ and $B$ fields vary sinusoidally in space and time. The average energy density $u$ of an electromagnetic wave can be obtained by replacing $E$ and $B$ with their rms values in the above equation. Thus

$$
\begin{aligned}
u & =\frac{1}{2} \varepsilon_0 E_{\text {rms }}^2+\frac{1}{2 \mu_0} B_{\mathrm{rms}}^2 \\
\text { or } \quad u & =\frac{1}{4} \varepsilon_0 E_0^2+\frac{1}{4 \mu_0} B_0^2\qquad\left[\because E_{\text {rms }}=\frac{E_0}{\sqrt{2}}, B_{\mathrm{rms}}=\frac{B_0}{\sqrt{2}}\right]
\end{aligned}
$$

Moreover, $E_0=c B_0$ and $c^2=\frac{1}{\mu_0 \varepsilon_0}$, therefore
$$
\begin{aligned}
u_E & =\frac{1}{4} \varepsilon_0 E_0^2=\frac{1}{4} \varepsilon_0\left(c B_0\right)^2 \\
& =\frac{1}{4} \varepsilon_0 \cdot \frac{B_0^2}{\mu_0 \varepsilon_0}=\frac{1}{4 \mu_0} B_0^2=u_B
\end{aligned}
$$
Hence in an electromagnetic wave, the average energy of the $E$ field equals the average energy density of the B field.
It may be noted that
$$
u=\frac{1}{4} \varepsilon_0 E_0^2+\frac{1}{4} \varepsilon_0 E_0^2=\frac{1}{2} \varepsilon_0 E_0^2=\varepsilon_0 E_{\mathrm{rms}}^2
$$
Also,
$$
u=\frac{1}{4 \mu_0} B_0^2+\frac{1}{4 \mu_0} B_0^2=\frac{1}{2 \mu_0} B_0^2=\frac{1}{\mu_0} B_{\mathrm{rms}}^2
$$

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Frequently Asked Questions – FAQs

What is the energy density of electromagnetic waves?

The energy density of an electromagnetic wave is a measure of how much energy is stored in a certain volume of space occupied by the wave.

What is the average energy density of electromagnetic wave class 12?

The average energy density of an electromagnetic wave is the time-averaged value of the energy density over a certain period of time.

Electromagnetic waves consist of oscillating electric and magnetic fields that carry energy through space, and this energy is distributed across the wave. The average energy density is calculated by taking the time average of the energy density at each point in space over a specific time interval.

Mathematically, the average energy density (u_avg) is given by:
$$u_{avg} = \frac{1}{T} \int_{t1}^{t2} u(t) dt$$

What is the formula of energy density Class 12?

The formula for the energy density (u) of an electromagnetic wave is given as:
$$u=\frac{1}{2} \varepsilon_0 E^2+\frac{1}{2 \mu_0} B^2$$

What is the maximum energy density of an electromagnetic wave?

The maximum energy density of an electromagnetic wave is not fixed and depends on the frequency and intensity of the wave. The energy density of an electromagnetic wave is proportional to the square of the electric and magnetic field strengths.

The maximum energy density of an electromagnetic wave is reached when the electric and magnetic fields are at their maximum values. For example, gamma rays, which have very high frequencies and intense fields, can have energy densities of billions of joules per cubic meter. In contrast, radio waves, which have lower frequencies and weaker fields, have much lower energy densities, typically on the order of microjoules per cubic meter or less.

So, the maximum energy density of an electromagnetic wave can vary widely depending on the frequency and intensity of the wave.

What is the symbol for energy density?

The symbol for energy density is small $u$.

Stay tuned with Laws Of Nature for more useful and interesting content.

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