Problem 67 - University Physics with Modern Physics

Question

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.

Step-by-Step Solution

Answer

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 1: 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: \(I=\dfrac{5.00 \times 10^{-3} W}{\dfrac{1}{4} \pi (2.00 \times 10^{-3} m)^2}\)

Step 2: 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: \(E_\text{rms} = \sqrt{\dfrac{2I}{c\epsilon_0}}\)

Step 3: 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.