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Physics · lesson 4 of 4

Electromagnetic waves and energy flow

In vacuum, plane electromagnetic waves have transverse electric and magnetic fields with E0=cB0. Both fields are perpendicular to propagation.

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Learn the ideas

Prerequisites: Vector calculus and sinusoidal waves.

Vacuum waves

In vacuum, plane electromagnetic waves have transverse electric and magnetic fields with E0=cB0. Both fields are perpendicular to propagation.

Energy flux

The Poynting vector is S=(E×B)/μ0 in vacuum. Its direction gives electromagnetic energy flow in this model.

Intensity

For a sinusoidal vacuum plane wave, mean intensity is ε0cE0²/2. Peak and rms field conventions must not be interchanged unnoticed.

Dispersion

Phase speed ω/k and group speed dω/dk need not coincide in a dispersive medium. Vacuum propagation is a special nondispersive case.

Worked examples

Example 1

In a vacuum plane wave, how are E and B oriented relative to propagation?

Maxwell plane-wave solutions have mutually perpendicular E, B and propagation directions.

Both are transverse

Example 2

Use c=3×10^8 m/s. For B0=2×10^−6 T, find E0.

E0=cB0=600 V/m.

600 V/m

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 In a vacuum plane wave, how are E and B oriented relative to propagation?

Show answer and explanation

Both are transverse

Maxwell plane-wave solutions have mutually perpendicular E, B and propagation directions.

2 Use c=3×10^8 m/s. For B0=2×10^−6 T, find E0.V/m

Show answer and explanation

600 V/m

E0=cB0=600 V/m.

3 Which vector sets electromagnetic energy-flow direction in vacuum?

Show answer and explanation

E cross B

The Poynting vector is proportional to E×B.

4 If a sinusoidal electric-field amplitude doubles, by what factor does mean intensity change?

Show answer and explanation

4

Mean intensity is proportional to E0².

5 For dispersion relation ω=ak², which is group speed?

Show answer and explanation

2ak

Differentiate ω with respect to k.

6 For ω=2k² in consistent units at k=3, calculate group speed.m/s

Show answer and explanation

12 m/s

dω/dk=4k=12 in the stated units.

7 Explain why peak field cannot simply be substituted into a formula written for rms field.

Sample answer and marking guidance

For a sinusoid Erms=E0/sqrt(2). Using the wrong convention changes the squared field factor and hence the inferred mean intensity.

Credit the sqrt(2) relation and its intensity consequence.

8 Describe a right-hand-rule check on the propagation direction when E points +x and B points +y.

Sample answer and marking guidance

E×B points +z, so the vacuum plane wave carries energy along +z under this orientation.

Credit cross-product order, not B×E.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.