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Chapter 31: Problem 67

A \(5.00-\mathrm{mW}\) laser pointer has a beam diameter of \(2.00 \mathrm{~mm}\) a) What is the root-mean-square value of the electric field in this laser beam? b) Calculate the total electromagnetic energy in \(1.00 \mathrm{~m}\) of this laser beam.

Short Answer

Expert verified
Solution: Step 1: We calculate the intensity of the laser beam (I) using the given power (P) and diameter (d). We find the area (A) of the beam as: \(A = \dfrac{1}{4} \pi d^2\) Then, we calculate the intensity as: \(I=\dfrac{5.00 \times 10^{-3} W}{\dfrac{1}{4} \pi (2.00 \times 10^{-3} m)^2}\) Step 2: We find the root-mean-square value of the electric field (E_rms) using the Poynting vector (S) and the intensity (I): \(E_\text{rms} = \sqrt{\dfrac{2I}{c\epsilon_0}}\) Step 3: We calculate the total electromagnetic energy (U) in a 1.00-meter length of the laser beam using the calculated intensity (I) and area (A): \(U = I \cdot A \cdot L\) By completing the calculations and plugging in the values, we can find the root-mean-square value of the electric field and the total electromagnetic energy in the laser beam.

Step by step solution

01

Calculating the intensity of the laser beam

The intensity of the laser beam (\(I\)) can be calculated using the following expression: \(I = \dfrac{P}{A}\) Where \(P\) is the power and \(A\) is the cross-sectional area of the beam. We have the power value \(P = 5.00 \, \text{mW}\), and since the laser beam's diameter is given (\(d = 2.00 \, \text{mm}\)), we can find the area as follows: \(A = \dfrac{1}{4} \pi d^2\) The intensity is then: \(I=\dfrac{5.00 \times 10^{-3} W}{\dfrac{1}{4} \pi (2.00 \times 10^{-3} m)^2}\)
02

Finding the root-mean-square value of the electric field

The Poynting vector \(S\) represents the energy flow in an electromagnetic wave and its magnitude is given by: \(S = \dfrac{1}{2} c \epsilon_0 E_\text{rms}^2 = cB_\text{rms}^2\) Where \(c\) is the speed of light (\(3.00\times10^8 \, \text{m/s}\)), \(\epsilon_0\) is the vacuum permittivity (\(8.85\times10^{-12} \, C^2/Nm^2\)), and \(E_\text{rms}\) and \(B_\text{rms}\) are the root-mean-square values of the electric and magnetic fields, respectively. Since we're looking for the root-mean-square value of the electric field, we can rearrange this equation as: \(E_\text{rms} = \sqrt{\dfrac{2S}{c\epsilon_0}}\) We will substitute the intensity we found in step 1 as the magnitude of the Poynting vector \(S\) to find the root-mean-square value of the electric field: \(E_\text{rms} = \sqrt{\dfrac{2I}{c\epsilon_0}}\)
03

Calculating the total electromagnetic energy in the laser beam

To find the total electromagnetic energy in a distance of \(1.00 \, \text{m}\) along the laser beam, we can use the following relation: \(U = I \cdot A \cdot L\) Where \(I\) is the intensity of the laser beam, \(A\) is the cross-sectional area, and \(L = 1.00 \, \text{m}\) is the distance along the laser beam. Using the values of intensity and area calculated previously, we can find the total electromagnetic energy in the laser beam.

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Most popular questions from this chapter

At the surface of the Earth, the Sun delivers an estimated \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\) of energy. Suppose sunlight hits a \(10.0 \mathrm{~m}\) by \(30.0 \mathrm{~m}\) roof at an angle of \(90.0^{\circ}\) a) Estimate the total power incident on the roof. b) Find the radiation pressure on the roof.

Alice made a telephone call from her home telephone in New York to her fiancé stationed in Baghdad, about \(10,000 \mathrm{~km}\) away, and the signal was carried on a telephone cable. The following day, Alice called her fiancé again from work using her cell phone, and the signal was transmitted via a satellite \(36,000 \mathrm{~km}\) above the Earth's surface, halfway between New York and Baghdad. Estimate the time taken for the signals sent by (a) the telephone cable and (b) via the satellite to reach Baghdad, assuming that the signal speed in both cases is the same as speed of light, \(c .\) Would there be a noticeable delay in either case?

A dipole antenna is located at the origin with its axis along the \(z\) -axis. As electric current oscillates up and down the antenna, polarized electromagnetic radiation travels away from the antenna along the positive \(y\) -axis. What are the possible directions of electric and magnetic fields at point \(A\) on the \(y\) -axis? Explain.

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy \(E=\hbar \omega,\) where \(\hbar=1.054573 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}\) is Planck's reduced constant and \(\omega\) is the angular frequency of the wave. a) Find the momentum of a photon. b) Find the angular momentum of a photon. Photons are circularly polarized; that is, they are described by a superposition of two plane-polarized waves with equal field amplitudes, equal frequencies, and perpendicular polarizations, one-quarter of a cycle \(\left(90^{\circ}\right.\) or \(\pi / 2\) rad \()\) out of phase, so the electric and magnetic field vectors at any fixed point rotate in a circle with the angular frequency of the waves. It can be shown that a circularly polarized wave of energy \(U\) and angular frequency \(\omega\) has an angular momentum of magnitude \(L=U / \omega .\) (The direction of the angular momentum is given by the thumb of the right hand, when the fingers are curled in the direction in which the field vectors circulate. c) The ratio of the angular momentum of a particle to \(\hbar\) is its spin quantum number. Determine the spin quantum number of the photon.

A 10.0 -mW vertically polarized laser beam passes through a polarizer whose polarizing angle is \(30.0^{\circ}\) from the horizontal. What is the power of the laser beam when it emerges from the polarizer?
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