Question

A laser produces light that is polarized in the vertical direction. The light travels in the positive \(y\) -direction and passes through two polarizers, which have polarizing angles of \(35.0^{\circ}\) and \(55.0^{\circ}\) from the vertical, as shown in the figure. The laser beam is collimated (neither converging nor expanding), has a circular cross section with a diameter of \(1.00 \mathrm{~mm}\), and has an average power of \(15.0 \mathrm{~mW}\) at point \(A\). At point \(C\), what are the amplitudes of the electric and magnetic fields, and what is the intensity of the laser light?

Short answer

Answer: At point C, the amplitudes of the electric and magnetic fields are approximately 3.005 x 10^3 N/C and 1.002 x 10^-5 T, respectively. The intensity of the laser light at point C is approximately 8.97 mW.

Step-by-step solution

Find the angle between the polarizers

Calculate the angle between the polarizers by finding the difference between their angles. \(\Delta \theta = 55.0^{\circ} - 35.0^{\circ} = 20^{\circ}\)

Determine the intensity at point C using Malus's law

Malus's law states that the intensity after passing through a polarizer is proportional to the square of the cosine of the angle between the incident light's polarization and the polarizer's axis. Since the light has to pass through two polarizers, we will apply Malus's law twice. \(I_{B} = I_{A} \cos^2{\left(35.0^{\circ}\right)}\) \(I_{C} = I_{B} \cos^2{\left(\Delta \theta\right)} = I_{A} \cos^2{\left(35.0^{\circ}\right)} \cos^2{\left(20.0^{\circ}\right)}\) Plug in the values: \(I_{C} = 15.0 \,mW \times \cos^2{\left(35.0^{\circ}\right)} \times \cos^2{\left(20.0^{\circ}\right)} \approx 8.97 \, mW\)

Determine the Poynting vector (S)

The Poynting vector (S) is related to the intensity I, with their relationship being: \(S = \frac{I}{c\mu_0}\) Where \(c \approx 3.0 \cdot 10^8 \, m/s\) is the speed of light, and \(\mu_0 \approx 4\pi \cdot 10^{-7} \, T\cdot m/A\) is the permeability of free space. Plug in the values to find S at point C: \(S_{C} = \frac{8.97 \cdot 10^{-3} \, W}{(3 \cdot 10^8 \, m/s) (4\pi \cdot 10^{-7} \, T\cdot m/A)} \approx 2.99 \cdot 10^4 \, N/C\)

Calculate the amplitudes of the electric (\(E\)) and magnetic (\(B\)) fields

Recall that the Poynting vector is related to the electric and magnetic fields by: \(S = \frac{1}{\mu_0} E B\) Now, we can isolate either the electric field or the magnetic field and find their amplitudes. Plugging in the values for point C, and solving for the amplitude of the electric field: \(E_{C} = \frac{S_{C} \mu_0}{B_C} = \frac{2.99 \cdot 10^4 \, N/C}{B_C} \implies E_{C}B_C = 2.99 \cdot 10^4 \, N/C\) In electromagnetic waves, the electric and magnetic fields are related by: \( E = cB \) We can replace \(E_{C}\) with \(c B_{C}\) in the equation \(E_{C}B_C = 2.99 \cdot 10^4 \, N/C\): \(c B_{C}^2 = 2.99 \cdot 10^4 \, N/C\) Now we can solve for the amplitude of the magnetic field: \(B_{C} = \sqrt{\frac{2.99 \cdot 10^4 \, N/C}{3 \cdot 10^8 \, m/s}} \approx 1.002 \cdot 10^{-5} \, T\) Substitute the value of \(B_C\) into the formula for electric field: \(E_{C} = cB_{C} = (3.0 \cdot 10^8 \, m/s)(1.002 \cdot 10^{-5} \, T) \approx 3.005 \cdot 10^{3} \, N/m\)

Final Results

At point C, the amplitudes of the electric and magnetic fields are: - Electric field amplitude: \(E_{C} \approx 3.005 \times 10^{3} \, N/C\) - Magnetic field amplitude: \(B_{C} \approx 1.002 \times 10^{-5} \, T\) The intensity of the laser light at point C is: - Intensity: \(I_{C} \approx 8.97 \, mW\)

Key concepts

Malus's Law

Malus's law plays a fundamental role in understanding the behavior of polarized light as it passes through polarizing filters. This law expresses the relationship between the intensity of the polarized light and the angle of the polarizer relative to the light's initial polarization direction. It is given by the equation:
\(I = I_0 \times \text{cos}^2(\theta)\)
where \(I\) is the transmitted intensity, \(I_0\) is the initial intensity, and \(\theta\) is the angle between the light's polarization direction and the axis of the polarizer.
In the given exercise, we saw the application of Malus's law, where light passes through two polarizers. When dealing with multiple polarizers, we apply Malus's Law sequentially for each polarizer. This is crucial because it tells us that the intensity of light is not simply diminished randomly; it follows a predictable pattern based on the polarizer's orientation. By calculating the intensity after each polarizer, students can gain a deeper understanding of how light interacts with materials that affect its polarization.

Poynting Vector

The Poynting vector is an essential concept in electromagnetism, representing the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. The magnitude of the Poynting vector is the power per unit area carried by the wave, which is what we commonly refer to as intensity in the context of light waves. The Poynting vector \(\vec{S}\) is defined by the following relationship:
\(\vec{S} = \frac{1}{\text{\(\mu_0\)}} \vec{E} \times \vec{B}\)
where \(\vec{E}\) is the electric field vector, \(\vec{B}\) is the magnetic field vector, and \(\mu_0\) is the permeability of free space.
In the context of our exercise, determining the Poynting vector's magnitude allowed us to find the values of the electric and magnetic field amplitudes. Understanding how the Poynting vector correlates to these field amplitudes and the intensity of light gives insight into how energy is transported in electromagnetic waves, such as the laser light in our example.

Electric and Magnetic Fields of Laser Light

Laser light, like any electromagnetic wave, is characterized by its electric and magnetic fields, which oscillate perpendicularly to each other and to the direction of the wave's propagation. In the context of lasers, these fields are particularly well-defined and coherent. The relationship between the electric field amplitude \(E\) and the magnetic field amplitude \(B\) in a vacuum is given by the equation:
\(E = cB\)
where \(c\) is the speed of light in a vacuum. This simple equation illustrates that the electric and magnetic field amplitudes are directly proportional to each other and provides a straightforward method for calculating one field if the other is known.
In the provided exercise, the use of this relationship, combined with the calculated Poynting vector, allowed us to solve for the amplitudes of both fields after the laser light passed through the polarizers. By understanding this relationship and the character of laser light, students learn how electromagnetic waves are structured and how energy is carried through space.