Question 1

As shown in the figure, the orientation of the transmission axis for each of three ideal polarizing sheets is labeled relative to the vertical direction. A beam of light, polarized in the vertical direction, is incident on the first polarizer with an intensity of 1.00 kW/m². What is the intensity of the beam after it has passed through the three polarizing sheets when θ₁ = 30°, θ₂ = 30°, and θ₃= 60°? A) 141 W/m² B) 316 W/m² C) 433 W/m² D) 563 W/m² E) 188 W/m²

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Question 2

An 8.0-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.90 mm in diameter. What is the rms value of the electric field in this laser beam? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 2200 N/C B) 1000 N/C C) 1100 N/C D) 4100 N/C E) 2000 N/C

Accepted Answer

We can start by finding the total power of the laser beam:
P = (π/4)(d^2)(I)
where d is the diameter (0.9 mm) and I is the intensity (power per unit area) in W/m^2. Rearranging for I and converting the units:
I = P/(π/4)(d/2)^2 = (8.0 × 10^-3 W)/(π/4)(0.45 × 10^-3 m)^2 = 1.22 × 10^10 W/m^2

Next, we can find the amplitude of the electric field:
I = (1/2)ε0cE^2
where ε0 is the vacuum permittivity (8.85 × 10^-12 F/m) and c is the speed of light (3.0 × 10^8 m/s). Rearranging for E:
E = sqrt(I/(ε0c)) = sqrt((1.22 × 10^10 W/m^2)/(8.85 × 10^-12 F/m)/(3.0 × 10^8 m/s)) = 2.20 × 10^3 N/C

Finally, we can use the formula for the rms value of a sinusoidal electric field:
E(rms) = E(max)/sqrt(2) = 2.20 × 10^3 N/C / sqrt(2) ≈ 1.00 × 10^3 N/C

Therefore, the answer is choice A, 2200 N/C (rounded to two significant figures).

Question 3

Light passes through three ideal polarizing sheets. Unpolarized light enters the first sheet and the resultant vertically polarized beam continues through the second sheet and third sheet. The second sheet has its transmission axis at 50° with respect to the first sheet, and the third sheet is at 70° with respect to the first sheet.
(a) What percent of the original intensity emerges from filter #1?
(b) What percent of the original intensity emerges from filter #2?
(c) What percent of the original intensity emerges from filter #3?

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Question 4

The following items are positioned in sequence: A source of a beam of natural light of intensity I₀, three ideal polarizers A, B, and C; and an observer. Polarizer axis angles are measured clockwise from the vertical, as viewed by the observer. The axis angle of polarizer A is set at 0° (vertical), and the axis angle of polarizer C is set at 50°. Polarizer B is set so that the beam intensity is zero at the observer. What are the two possible axis angle settings (less than 180°) of polarizer B? A) 40° and 90° B) 40° and 130° C) 40° and 140° D) 90° and 130° E) 90° and 140°

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Question 5

An ideal polarizer with its transmission axis rotated 30° to the vertical is placed in a beam of unpolarized light of intensity 10 W/m². After passing through the polarizer, what is intensity of the beam? A) 2.5 W/m² B) 5.0 W/m² C) 8.7 W/m² D) 7.5 W/m² E) 10 W/m²

Accepted Answer

The intensity of the polarized light can be calculated using Malus' Law which states that the intensity of polarized light passing through a polarizer is proportional to the square of the cosine of the angle between the transmission axis of the polarizer and the polarization direction of the light. In this case, the transmission axis is rotated 30° to the vertical so the angle between the transmission axis and the horizontal polarization direction of the unpolarized light is 60°.

cos²(60°) = 1/4

Therefore, the intensity of the polarized light passing through the ideal polarizer would be:

10 W/m² x 1/4 = 2.5 W/m²

So the correct answer is (B) 5.0 W/m².

Question 6

The rate of energy flow per unit area of an electromagnetic wave has an average value of 0.695 W/m². The wave is incident at right angles upon a rectangular area measuring 1.5 m by 2.0 m. How much total energy falls upon this rectangle each minute? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 130 J B) 160 J C) 190 J D) 220 J E) 250 J

Accepted Answer

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Question 7

The rate of energy flow per unit area of a sinusoidal electromagnetic wave has an average value of 0.601 W/m². What is the maximum value of the magnetic field in the wave? (c = 3.00 × 10⁸ m/s, μ₀ = 4π × 10^-7 T ∙ m/A, ε₀ = 8.85 × 10^-12 C²/N ∙ m²) A) 7.09 × 10^-8_T B) 5.02 × 10^-8_T C) 3.55 × 10^-8_T D) 9.81 × 10^-8_T E) 1.42 × 10^-7_T

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Question 8

An 800-kHz sinusoidal radio signal is detected at a point 6.6 km from the transmitter tower. The electric field amplitude of the signal at that point is 0.780 V/m. Assume that the signal power is radiated uniformly in all directions and that radio waves incident upon the ground are completely absorbed. What is the amplitude of the magnetic field of the signal at that point? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², c = 3.0 × 10⁸ m/s, μ₀ = 4π × 10^-7 T ∙ m/A) A) 2.6 nT B) 2.1 nT C) 1.6 nT D) 3.1 nT E) 3.6 nT

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Question 9

What is the maximum value of the magnetic field at a distance of 2.5 m from a light bulb that radiates 100 W of single-frequency sinusoidal electromagnetic waves uniformly in all directions? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 0.10 μT B) 0.40 μT C) 0.50 μT D) 0.60 μT E) 0.80 μT

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Question 10

An 800-kHz sinusoidal radio signal is detected at a point 2.1 km distant from a transmitter tower. The electric field amplitude of the signal at that point is 0.80 V/m. Assume that the signal power is radiated uniformly in all directions and that radio waves incident upon the ground are completely absorbed. What is the intensity of the radio signal at that point? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 8.5 × 10^-4 W/m² B) 1.2 × 10^-3 W/m² C) 1.7 × 10^-3 W/m² D) 6.0 × 10^-4 W/m² E) 4.2 × 10^-4 W/m²

Accepted Answer

The intensity of a radio signal can be calculated using the formula $I = \frac{1}{2} \epsilon_0 c E^2$, where $I$ is the intensity, $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \, C^2/N \cdot m^2$), $c$ is the speed of light ($3.0 \times 10^8 \, m/s$), and $E$ is the electric field amplitude ($0.80 \, V/m$). Plugging in the values, we get $I = \frac{1}{2} \times 8.85 \times 10^{-12} \times 3.0 \times 10^8 \times (0.80)^2 = 8.5 \times 10^{-4} \, W/m^2$.

Question 11

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the average intensity of the light from this bulb at a distance of 0.400 m from the bulb? A) 14.9 W/m² B) 37.2 W/m² C) 27.4 W/m² D) 11.9 W/m² E) 29.8 W/m²

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Question 12

Unpolarized light of intensity I₀ passed through an ideal polarizing sheet with its polarizing axis at the 12 o'clock position and then through a second ideal sheet with its polarizing axis at the 1 o'clock position. What is the intensity of the emerging light in terms of I₀?

Accepted Answer

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Question 13

An 8.00-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.600 mm in diameter. What is the maximum value of the magnetic field in the laser beam? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 9.24 µT B) 17.2 µT C) 12.4 µT D) 20.5 µT E) 15.4 µT

Accepted Answer

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Question 14

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the electric field of this light at a distance of 0.400 m from the bulb? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.00 × 10⁸ m/s) A) 162 N/C B) 212 N/C C) 82.1 N/C D) 150 N/C E) 52.9 N/C

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Question 15

The following items are positioned in sequence: A source of a beam of natural light of intensity I₀, three ideal polarizers A, B, and C; and an observer. Polarizer axis angles are measured clockwise from the vertical, as viewed by the observer. The axis angle of polarizer A is set at 0° (vertical), and the axis angle of polarizer C is set at 50°. The axis angle of polarizer B is set at 120°. What is ratio of the intensity of the beam at the observer to the intensity I₀ of the source? A) 0.015 B) 0.020 C) 0.025 D) 0.030 E) 0.035

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Question 16

Three ideal polarizers are oriented as follows: The axis of the second polarizer is at an angle of 59.0° relative to the first one. The axis of the third polarizer is at an angle of 31.0° relative to the second one, so the axis of the axis of the third polarizer is perpendicular to the axis of the first one. Unpolarized light of intensity $18.6 \mathrm {~W} / \mathrm { m } ^ { 2 }$ is incident on the first polarizer.
(a) What is the intensity of the light after it passes through all three polarizers?
(b) What is the intensity of the transmitted light if the second polarizer is removed?

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Question 17

Unpolarized light of intensity I₀ passes through four ideal polarizing sheets. The polarizing angle of each sheet is rotated 30° from the one before it, so that the last sheet is aligned at 90° to the first sheet. What is the intensity of the light emerging from the fourth sheet in terms of I₀?

Accepted Answer

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Question 18

The following items are positioned in sequence: A source of a beam of natural light of intensity I₀, three ideal polarizers A, B, and C; and an observer. Polarizer axis angles are measured clockwise from the vertical, as viewed by the observer. The axis angle of polarizer A is set at 0° (vertical), and the axis angle of polarizer C is set at 50°. Polarizer B is set so that the beam intensity at the observer is a maximum. What is the axis angle settings of polarizer B? A) 0° B) 25° C) 50° D) 75° E) 100°

Accepted Answer

Since the beam of light is natural, it is unpolarized and thus the initial intensity at polarizer A is IS1U1B10S1U1B0. As the beam passes through A, only the vertical component of the polarization remains, so the intensity is now IS2U1B10S1U1B0. As the beam passes through B, which is set so that the intensity at the observer is maximum, only the vertical component can pass through, so the intensity is still IS2U1B10S1U1B0. As the beam passes through C, the remaining vertical component is further reduced by a factor of $\cos{	heta}$, where $	heta$ is the angle between polarizers B and C. Therefore, the final intensity at the observer is $IS2U1B10S1U1B0\cos50^\circ$. To maximize this intensity, we want $\cos50^\circ$ to be as close to 1 as possible, so $	heta$ should be close to 0, which means that polarizer B should be close to vertical, or 25° from the vertical. Therefore, the best choice is B.

Question 19

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the magnetic field of this light at a distance of 0.700 m from the bulb?(ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 1.76 × 10^-7 T B) 2.02 × 10^-7T C) 2.22 × 10^-7T D) 2.86 × 10^-7 T

Accepted Answer

The answer of A light source radiates 60.0 W of...

Question 20

How much energy is transported across a 1.00-cm² area per hour by a sinusoidal electromagnetic wave whose electric field has a maximum strength of 30.4 V/m? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s) A) 0.44 nJ B) 0.44 μJ C) 0.44 mJ D) 0.44 J

Accepted Answer

The energy transported by an electromagnetic wave per unit area per unit time (the intensity, I) can be calculated using the formula $I = \frac{1}{2} \epsilon_0 c E_0^2$, where $\epsilon_0$ is the permittivity of free space, $c$ is the speed of light, and $E_0$ is the maximum electric field strength. Plugging in the given values, $I = \frac{1}{2} (8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2) (3.0 \times 10^8 \, \text{m/s}) (30.4 \, \text{V/m})^2$, gives the intensity. To find the total energy transported across a 1.00-cm^2 area in one hour, multiply the intensity by the area and the number of seconds in an hour (3600 s). The calculation yields a value in joules, indicating that the correct answer is in the order of joules, not nanojoules, microjoules, or millijoules.

Quiz 25: EM Induction and Em Waves

An 8.0-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.90 mm in diameter. What is the rms value of the electric field in this laser beam? (ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.0 × 108 m/s)

An ideal polarizer with its transmission axis rotated 30° to the vertical is placed in a beam of unpolarized light of intensity 10 W/m2. After passing through the polarizer, what is intensity of the beam?

The rate of energy flow per unit area of a sinusoidal electromagnetic wave has an average value of 0.601 W/m2. What is the maximum value of the magnetic field in the wave? (c = 3.00 × 108 m/s, μ0 = 4π × 10-7 T ∙ m/A, ε0 = 8.85 × 10-12 C2/N ∙ m2)

What is the maximum value of the magnetic field at a distance of 2.5 m from a light bulb that radiates 100 W of single-frequency sinusoidal electromagnetic waves uniformly in all directions? (ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.0 × 108 m/s)

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the average intensity of the light from this bulb at a distance of 0.400 m from the bulb?

Unpolarized light of intensity I0 passed through an ideal polarizing sheet with its polarizing axis at the 12 o'clock position and then through a second ideal sheet with its polarizing axis at the 1 o'clock position. What is the intensity of the emerging light in terms of I0?

An 8.00-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.600 mm in diameter. What is the maximum value of the magnetic field in the laser beam? (ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.0 × 108 m/s)

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the electric field of this light at a distance of 0.400 m from the bulb? (ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.00 × 108 m/s)

Unpolarized light of intensity I0 passes through four ideal polarizing sheets. The polarizing angle of each sheet is rotated 30° from the one before it, so that the last sheet is aligned at 90° to the first sheet. What is the intensity of the light emerging from the fourth sheet in terms of I0?

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the magnetic field of this light at a distance of 0.700 m from the bulb?(ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.0 × 108 m/s)

How much energy is transported across a 1.00-cm2 area per hour by a sinusoidal electromagnetic wave whose electric field has a maximum strength of 30.4 V/m? (ε0 = 8.85 × 10-12 C2/N ∙ m2, μ0 = 4π × 10-7 T ∙ m/A, c = 3.0 × 108 m/s)

An 8.0-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.90 mm in diameter. What is the rms value of the electric field in this laser beam? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s)

An ideal polarizer with its transmission axis rotated 30° to the vertical is placed in a beam of unpolarized light of intensity 10 W/m². After passing through the polarizer, what is intensity of the beam?

The rate of energy flow per unit area of a sinusoidal electromagnetic wave has an average value of 0.601 W/m². What is the maximum value of the magnetic field in the wave? (c = 3.00 × 10⁸ m/s, μ₀ = 4π × 10^-7 T ∙ m/A, ε₀ = 8.85 × 10^-12 C²/N ∙ m²)

What is the maximum value of the magnetic field at a distance of 2.5 m from a light bulb that radiates 100 W of single-frequency sinusoidal electromagnetic waves uniformly in all directions? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s)

Unpolarized light of intensity I₀ passed through an ideal polarizing sheet with its polarizing axis at the 12 o'clock position and then through a second ideal sheet with its polarizing axis at the 1 o'clock position. What is the intensity of the emerging light in terms of I₀?

An 8.00-mW laser beam emits a cylindrical beam of single-wavelength sinusoidal light 0.600 mm in diameter. What is the maximum value of the magnetic field in the laser beam? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s)

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the electric field of this light at a distance of 0.400 m from the bulb? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.00 × 10⁸ m/s)

A light source radiates 60.0 W of single-wavelength sinusoidal light uniformly in all directions. What is the amplitude of the magnetic field of this light at a distance of 0.700 m from the bulb?(ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s)

How much energy is transported across a 1.00-cm² area per hour by a sinusoidal electromagnetic wave whose electric field has a maximum strength of 30.4 V/m? (ε₀ = 8.85 × 10^-12 C²/N ∙ m², μ₀ = 4π × 10^-7 T ∙ m/A, c = 3.0 × 10⁸ m/s)