# 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.

Course: [Fields, circuits and thermal physics](https://studybrix.com/en/courses/fields-and-thermal-physics/index.md)
Level: university
Prerequisites: Vector calculus and sinusoidal waves.

## Notes

### 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.

## Practice questions

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

A. Both are transverse
B. Only B exists
C. Only E exists
D. Both are parallel to propagation

Answer: Both are transverse

Explanation: 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.

Answer: 600 V/m

Explanation: E0=cB0=600 V/m.

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

A. Position alone
B. E minus B
C. E cross B
D. E plus B without units

Answer: E cross B

Explanation: The Poynting vector is proportional to E×B.

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

Answer: 4

Explanation: Mean intensity is proportional to E0².

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

A. 2ak
B. ak
C. a/k
D. Zero for all k

Answer: 2ak

Explanation: Differentiate ω with respect to k.

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

Answer: 12 m/s

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

Explanation: 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 / marking guidance: E×B points +z, so the vacuum plane wave carries energy along +z under this orientation.

Explanation: Credit cross-product order, not B×E.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.