∙Predict/Calculate (a) Find the electric and magnetic field...

$\bullet$ Predict/Calculate (a) Find the electric and magnetic field amplitudes in an electromagnetic wave that has an average energy density of 1.0 $\mathrm{J} / \mathrm{m}^{3} .$ (b) By what factor must the field amplitudes be increased if the average energy density is to be doubled to 2.0 $\mathrm{J} / \mathrm{m}^{3} ?$

$\bullet$ Predict/Calculate (a) Find the electric and magnetic field
amplitudes in an electromagnetic wave that has an average energy
density of 1.0 $\mathrm{J} / \mathrm{m}^{3} .$ (b) By what factor must the field amplitudes be
increased if the average energy density is to be doubled to 2.0 $\mathrm{J} / \mathrm{m}^{3} ?$

Key Concepts

Relationship Between Field Amplitudes and Energy Density

The energy density in electromagnetic waves is directly proportional to the square of the field amplitudes (E and B). This quadratic dependence implies that any change in energy density requires a corresponding proportional change in the field amplitudes, following a square root relationship. This is crucial when determining how much the field amplitudes must be increased to achieve a specified change in energy density.

Scaling of Field Amplitudes with Energy Density

Because the energy density depends on the square of the field amplitudes, if the energy density is doubled, the field amplitudes do not double but are increased by a factor equal to the square root of the increase. This scaling behavior is a fundamental aspect in the analysis and design of electromagnetic systems where energy control is necessary.

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space. In a vacuum, these fields are perpendicular to one another and to the direction of propagation, and they maintain a fixed ratio of magnitudes (E = cB, where c is the speed of light). This intrinsic relationship is essential to understanding how energy is transported in such waves.

Energy Density in Electromagnetic Fields

The energy density of an electromagnetic field comprises contributions from both the electric and magnetic fields. In free space, the energy density is given by u = (1/2)??E² + (1/2??)B². For propagating electromagnetic waves, these two contributions are equal when averaged over time, which simplifies energy calculations for the wave.

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