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Speed, Energy Density, and Momentum of Electromagnetic Waves

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Student-friendly explanation

How to write this in exams

1
Start with the exact idea
The speed of electromagnetic waves in vacuum is c = 1/sqrt(mu_0 epsilon_0), approximately 3.0 x 10^8 m s^-1. Electromagnetic waves carry energy and momentum through their electric and magnetic fields.
2
Then show how to use it
1. Identify the required quantity: speed, energy density, or momentum. 2. Select the correct formula. 3. Convert all values to SI units. 4. Substitute carefully with powers of ten. 5. State the final answer with unit and physical meaning.
3
Add one concrete example
Sunlight carries energy from the Sun to Earth. A small force due to radiation pressure is exerted when light falls on a surface, though it is usually too small to notice in daily life.
4
Avoid this incomplete answer
A common wrong answer is p = U x c. The correct relation is p = U/c for electromagnetic radiation in vacuum.

Definition

Example

Sunlight carries energy from the Sun to Earth. A small force due to radiation pressure is exerted when light falls on a surface, though it is usually too small to notice in daily life.

Rule to remember

Speed: c = 1/sqrt(mu_0 epsilon_0), where mu_0 is permeability of free space in N A^-2 and epsilon_0 is permittivity of free space in F m^-1. Energy density: u = (1/2)epsilon_0 E^2 + B^2/(2mu_0), where u is in J m^-3, E is in V m^-1, and B is in tesla. Momentum relation: p = U/c for energy U carried by radiation in vacuum.

Memory hook

Light is fast because vacuum constants set c; light is not massless in effect because it still carries momentum.

Examples and method

Worked example

Find the momentum carried by electromagnetic radiation of energy 6.0 x 10^-3 J in vacuum. Use p = U/c = (6.0 x 10^-3)/(3.0 x 10^8) = 2.0 x 10^-11 kg m s^-1. Interpretation: even a small light pulse carries momentum, but the value is very small because c is large.

Method to apply

1. Identify the required quantity: speed, energy density, or momentum. 2. Select the correct formula. 3. Convert all values to SI units. 4. Substitute carefully with powers of ten. 5. State the final answer with unit and physical meaning.

Diagram support

A diagram is optional. If used, show a plane wave carrying energy in the propagation direction and label E, B, and energy flow direction. For numerical focus, a formula box is more useful than a detailed diagram.

How CBSE asks it

Questions may ask for deriving or using c = 1/sqrt(mu_0 epsilon_0), calculating energy density, comparing electric and magnetic energy parts, or finding radiation momentum from energy.

Avoid common mistakes

Common confusion

A frequent error is using E and B without SI units or forgetting that the magnetic energy term is B^2/(2mu_0), not B^2/(2epsilon_0).

Common wrong answer

A common wrong answer is p = U x c. The correct relation is p = U/c for electromagnetic radiation in vacuum.

Exam tip

In numerical answers, write units at every substitution step, especially for c, energy density, and momentum. This prevents mixing electric and magnetic field terms incorrectly.

Quick check

What does c = 1/sqrt(mu_0 epsilon_0) show about electromagnetic waves?

It shows that the speed of electromagnetic waves in vacuum is determined by the electric and magnetic constants of free space. Substituting mu_0 and epsilon_0 gives about 3.0 x 10^8 m s^-1, the speed of light.

Answer writing and exam use

1-mark answer

2-mark answer

The speed of electromagnetic waves in vacuum is c = 1/sqrt(mu_0 epsilon_0), approximately 3.0 x 10^8 m s^-1. Electromagnetic waves carry energy and momentum through their electric and magnetic fields. Speed: c = 1/sqrt(mu_0 epsilon_0), where mu_0 is permeability of free space in N A^-2 and epsilon_0 is permittivity of free space in F m^-1. Energy density: u = (1/2)epsilon_0 E^2 + B^2/(2mu_0), where u is in J m^-3, E is in V m^-1, and B is in tesla. Momentum relation: p = U/c for energy U carried by radiation in vacuum. Sunlight carries energy from the Sun to Earth. A small force due to radiation pressure is exerted when light falls on a surface, though it is usually too small to notice in daily life.

3-mark answer

The electric and magnetic fields in an electromagnetic wave store energy. The total energy density is the sum of electric field energy density and magnetic field energy density. In vacuum, these contributions are equal on average for a plane wave. Since electromagnetic waves carry momentum, radiation can exert pressure when absorbed or reflected. Speed: c = 1/sqrt(mu_0 epsilon_0), where mu_0 is permeability of free space in N A^-2 and epsilon_0 is permittivity of free space in F m^-1. Energy density: u = (1/2)epsilon_0 E^2 + B^2/(2mu_0), where u is in J m^-3, E is in V m^-1, and B is in tesla. Momentum relation: p = U/c for energy U carried by radiation in vacuum. Find the momentum carried by electromagnetic radiation of energy 6.0 x 10^-3 J in vacuum. Use p = U/c = (6.0 x 10^-3)/(3.0 x 10^8) = 2.0 x 10^-11 kg m s^-1. Interpretation: even a small light pulse carries momentum, but the value is very small because c is large. Questions may ask for deriving or using c = 1/sqrt(mu_0 epsilon_0), calculating energy density, comparing electric and magnetic energy parts, or finding radiation momentum from energy. A common wrong answer is p = U x c. The correct relation is p = U/c for electromagnetic radiation in vacuum.

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