A small helium-neon laser emits red visible light with a...

A small helium-neon laser emits red visible light with a power of 3.20 $\mathrm{mW}$ in a beam that has a diameter of 2.50 $\mathrm{mm}$ . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a $1.00-\mathrm{m}$ length of the beam?

Key Concepts

Volume Integration for Energy Calculation

The total energy contained in an electromagnetic beam can be calculated by integrating the energy density over the volume that the beam occupies. For a beam with a known cross-sectional area and length, multiplying the energy density by that volume provides the total energy carried by the beam.

Energy Density in an Electromagnetic Field

Energy density represents the energy stored per unit volume within an electromagnetic field. It is calculated separately for the electric field (u_E = (1/2)??E²) and the magnetic field (u_B = B²/(2??)); in electromagnetic waves these densities are equal. This concept is key for understanding how energy is distributed in space within the wave.

Intensity of Electromagnetic Radiation

Intensity is defined as the power transmitted per unit area in an electromagnetic wave. It is directly related to the amplitude of the electric field by the expression I = (1/2)c??E², where c is the speed of light and ?? is the permittivity of free space. This relation connects the energy flow with the field’s amplitude.

Relationship between Electric and Magnetic Fields

In free space, the electric and magnetic fields in an electromagnetic wave are linked by the relationship E = cB, where c denotes the speed of light. This connection implies that if one field amplitude is known, the other can be directly determined, ensuring both fields contribute equally to the energy in the wave.

Electromagnetic Waves

Electromagnetic waves are self-propagating oscillations of electric and magnetic fields that travel through space. They are characterized by their frequency, wavelength, and speed (the speed of light in vacuum), and their behavior is governed by Maxwell’s equations.

Transcript

0:00 Hello everyone.

00:01 In this problem, we are given some basic information about a helium neon laser.

00:06 We're told that it has a power of 3 .20 times 10 through minus 3 watts, and it radiates in a diameter of 2 .50 times 10 through minus 3 meters.

00:20 And we're also later on going to find the energy contained in 1 meter of this laser beam.

00:26 So, but if we ever get to that, the first part is asking us to find what the amplitude of the electric magnetic fields are in this laser.

00:37 And so what we can do is, since we know the diameter of the laser beam and we know its power, we can calculate its intensity.

00:46 And its intensity is given by the power divided by the area over which the electromagnetic wave is radiated.

00:52 And so, of course, that's just the circle.

00:55 So if you imagine the laser beam, so this is your laser or something.

00:59 And it's radiating something like this.

01:06 And so this is the area over which it radies the electromagnetic waves.

01:12 Okay, so that's a circle.

01:14 So then for the area, we're going to use pi times d over two, because we're given the diameter, so d over two is the radius, so pi times the over two squared.

01:26 And then that gives us four times b over pi times d squared.

01:30 And we know that the intensity is also given in terms of the electromagnetic field amplitude, for example, as a half times epsilon zero times c times e squared.

01:41 So then just rearranging this equation, we find this relation.

01:47 So just again to emphasize, this is what we know, right? on the one hand side, we just use that the intensity is p over a, and then the other we use that it's half times epsilon zero times c times e squared.

02:01 And so just rearranging this for e squared, or for e, we find that e is 8 times p over at pi times epsilon 0 times c times d squared, all square rooted.

02:13 And then just plugging in the values, we find that the electric field amplitude is 700 .7 volts per meters...

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