Derive the formula for Intensity of electromagnetic wave, class 12

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In this short piece of article, we will derive a formula for the Intensity of electromagnetic waves. Going through this article, you will be able to derive the formula for the Intensity of electromagnetic waves by yourself. So, stay tuned with this article.

Intensity of electromagnetic wave formula derivation

The intensity of an electromagnetic wave is defined as the amount of energy passing through a unit area per unit of time.

Derive the formula for Intensity of electromagnetic wave, class 12
The intensity of electromagnetic wave formula derivation, source: Physics Stack Exchange

To derive the formula for intensity, we can start with the expression for the energy density of an electromagnetic wave.

The energy density u of an electromagnetic wave is given by:

\begin{equation} u = \frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0}B^2 \end{equation}

where E is the electric field, B is the magnetic field, ε0 is the permittivity of free space, and μ0 is the permeability of free space.

The time-averaged power per unit area carried by the wave is given by the Poynting vector, which is defined as:

\begin{equation} \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \end{equation}

where the cross product gives the direction of energy flow. The magnitude of the Poynting vector is the intensity I of the wave:

\begin{equation} I = |\vec{S}| = \frac{1}{\mu_0} |\vec{E}| |\vec{B}| \sin \theta \end{equation}

where θ is the angle between the electric and magnetic fields.

For a plane wave, the electric and magnetic fields are perpendicular to each other and to the direction of propagation, so θ is 90 degrees and sin θ is equal to 1. Thus, we have:

\begin{equation} I = \frac{1}{\mu_0} |\vec{E}| |\vec{B}| \end{equation}

Using the relationship between the electric and magnetic fields in a plane wave, namely E = cB, where c is the speed of light, we can simplify this expression to:

\begin{equation} I = \frac{1}{\mu_0 c} E^2 \end{equation}

This is the formula for the intensity of an electromagnetic wave in terms of the electric field strength E. It shows that the intensity is proportional to the square of the electric field amplitude and inversely proportional to the permeability of free space and the speed of light.

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Frequently Asked Questions – FAQs

What is the formula for the intensity of electromagnetic waves?

The intensity of an electromagnetic wave is a measure of the rate at which energy is transported by the wave per unit area perpendicular to the direction of wave propagation. Mathematically, it is defined as the power per unit area and is given by the formula:
\begin{equation} \text{Intensity} = \frac{\text{Power}}{\text{Area}} \end{equation}

What is the intensity of a wave?

The intensity of an electromagnetic wave is the rate at which energy is transported by the wave per unit area perpendicular to the direction of wave propagation

What is the intensity of electromagnetic waves in terms of the magnetic field?

The intensity of electromagnetic waves in terms of the magnetic field is given as: \begin{equation} I = \frac{1}{\mu_0 } cB^2 \end{equation}

What is the intensity of electromagnetic waves in terms of the electric field?

The intensity of electromagnetic waves in terms of the electric field is given as: \begin{equation} I = \frac{1}{\mu_0 c} E^2 \end{equation}

What is the unit of intensity?

The unit of intensity is watts per square meter (W/m²). 

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