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You want to support a sheet of fireproof paper horizontally, using only a vertical upward beam of light spread uniformly over the sheet. There is no other light on this paper. The sheet measures \(22.0 \mathrm{~cm}\) by \(28.0 \mathrm{~cm}\) and has a mass of \(1.50 \mathrm{~g}\). (a) If the paper is black and hence absorbs all the light that hits it, what must be the intensity of the light beam? (b) For the light in part (a), what are the maximum values of its electric and magnetic fields? (c) If the paper is white and hence reflects all the light that hits it, what intensity of light beam is needed to support it? (d) To see if it is physically reasonable to expect to support a sheet of paper this way, calculate the intensity in a typical \(0.500 \mathrm{~mW}\) laser beam that is \(1.00 \mathrm{~mm}\) in diameter and compare this value with your answer in part (a).

Step by step solution

01

Understanding Intensity Required

To support the paper, the light's upward force must balance the paper's weight. The force from light on absorbing material is given by the equation \( F = I \times A / c \), where \( I \) is the intensity, \( A \) is the area, and \( c \) is the speed of light. First, calculate the weight \( W \) of the paper: \( W = m \times g = 1.50 \times 10^{-3} \times 9.8 \). Next, calculate the area \( A = 0.22 \times 0.28 \). Set \( F = W \) to find \( I \).

02

Calculate Weight and Area

Calculate the weight: \( W = 1.50 \times 10^{-3} \times 9.8 = 1.47 \times 10^{-2} \mathrm{~N} \). Calculate the area: \( A = 0.22 \times 0.28 = 6.16 \times 10^{-2} \mathrm{~m}^2 \).

03

Solving for Intensity for Absorbing Paper

Set the force of the light equal to the weight: \( F = I \times A / c = W \). Rearrange for intensity: \( I = W \times c / A \). Substitute \( c = 3 \times 10^8 \mathrm{~m/s} \), \( W = 1.47 \times 10^{-2} \mathrm{~N} \), \( A = 6.16 \times 10^{-2} \mathrm{~m^2} \). Calculate \( I. \)

04

Determine Maximum Electric and Magnetic Fields

For the maximum values of electric and magnetic fields, use \( E = \sqrt{2I/c\varepsilon_0} \) and \( B = E/c \). Use values \( \varepsilon_0 = 8.85 \times 10^{-12} \mathrm{~F/m} \), \( I \) from Step 3, and \( c = 3 \times 10^8 \mathrm{~m/s}. \)

05

Solving for Intensity for Reflective Paper

For reflective paper, the momentum change doubles, so \( F = 2I \times A / c = W \). Thus, \( I = W \times c / (2A) \). Substitute \( W \) and \( A \) from Step 2.

06

Evaluate Laser Intensity

A typical laser beam power is given as \( P = 0.500 \mathrm{~mW} = 0.500 \times 10^{-3} \mathrm{~W} \) and the diameter as \( 1.00 \mathrm{~mm} \). Calculate the area \( A_\text{laser} = \pi \times (0.5 \times 10^{-3})^2 \). Then, intensity \( I_\text{laser} = P / A_\text{laser} \). Compare this with intensity from Step 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Radiation

Electromagnetic radiation is energy that travels through space as waves. These waves are created by oscillating electric and magnetic fields and can travel through a vacuum. This is how light is able to reach us from the Sun across space without needing a medium.

• Light, a form of electromagnetic radiation, is typically characterized by its wavelength or frequency.
• The entire electromagnetic spectrum includes radio waves, microwaves, infrared waves, visible light, ultraviolet rays, X-rays, and gamma rays.
• Visible light is just a small part of this spectrum, and it consists of the colors that the human eye can see.

Understanding electromagnetic radiation is crucial to fields like optics, where we examine how light interacts with different surfaces and mediums, such as the paper in our original exercise.

Pressure from Light

When light hits a surface, it exerts pressure, a concept that might seem strange at first since light is often perceived as intangible.

• This pressure comes from the momentum of the light photons.
• When photons strike a surface, they transfer momentum, which can create a force capable of moving objects as light or delicate as a sheet of paper.

The formula for calculating the pressure exerted by light involves the intensity of the light and the area it hits. In our exercise, if the paper absorbs all light, we use the formula \( F = \frac{I \times A}{c} \) to find the force needed to balance the paper's weight. When the paper reflects light, this force is doubled because the momentum change is effectively twice as much.

Photon Momentum

Photons, though massless, have momentum because they are packets of energy traveling at the speed of light. This can be understood better with Max Planck’s equation for energy of a photon, \( E = hf \).

• Here, \( h \) is the Planck constant and \( f \) is the frequency of the light.
• Since momentum \( p \) is defined differently for photons than classical particles, it uses the relation \( p = \frac{E}{c} \).

This momentum is key in calculating the effect of light pressures explored in previous sections. By understanding that momentum is conserved, we figure out how light could theoretically move or support objects, like the sheets of paper in our problem, through its impact and subsequent transfer of energy.

Energy and Power in Optics

In the realm of optics, energy and power are essential considerations when analyzing light's interaction with materials.

• Intensity, a measure often used in optics, refers to the power per unit area carried by a wave. It is expressed in watts per square meter (W/m²).
• The power of a light source, like a laser, is the rate at which it emits energy, measured in watts (W).

The intensity needed to support an object, such as in the original exercise, can be calculated by considering the balance between the gravitational force and the force of the light. By understanding the relationship between power, intensity, and area, students can appreciate how different light sources compare, such as the difference between the calculated intensity for supporting a paper and a typical laser beam.