medium difficulty9 min

Speed, Energy Density, and Momentum of Electromagnetic Waves

The speed of electromagnetic waves in vacuum is c = 1/√(μ₀ ε₀), approximately 3.0 × 10⁸ m s⁻¹. Electromagnetic waves carry energy and momentum through their electric and magnetic fields.

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

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Understand it

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.

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.

Formula and rule

Speed: c = 1/√(μ₀ ε₀), where μ₀ is permeability of free space in N A⁻² and ε₀ is permittivity of free space in F m⁻¹.

Energy density: u = (1/2)ε₀ E² + B²/(2μ₀), 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.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the momentum carried by electromagnetic radiation of energy 6.0 × 10⁻³ J in vacuum.

1Use p = U/c = (6.0 × 10⁻³)/(3.0 × 10⁸) = 2.0 × 10⁻¹¹ kg m s⁻¹.
2Interpretation: even a small light pulse carries momentum, but the value is very small because c is large.

Method you can reuse

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

Visual tip

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.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does c = 1/√(μ₀ ε₀) show about electromagnetic waves?

Think first, then show answer

It shows that the speed of electromagnetic waves in vacuum is determined by the electric and magnetic constants of free space. Substituting μ₀ and ε₀ gives about 3.0 × 10⁸ m s⁻¹, the speed of light.

Common wrong answer

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

Don't mix this up

A frequent error is using E and B without SI units or forgetting that the magnetic energy term is B²/(2μ₀), not B²/(2ε₀).

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.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask for deriving or using c = 1/√(μ₀ ε₀), calculating energy density, comparing electric and magnetic energy parts, or finding radiation momentum from energy.

1
Enough for 1 mark
The speed of electromagnetic waves in vacuum is c = 1/√(μ₀ ε₀), approximately 3.0 × 10⁸ m s⁻¹. Electromagnetic waves carry energy and momentum through their electric and magnetic fields.
2
Add for 2 marks
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.
3
Add for 3 marks
The electric and magnetic fields in an electromagnetic wave store energy.

Practice

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