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Chapter 34: Problem 24

A black, totally absorbing piece of cardboard of area \(A=2.0 \mathrm{~cm}^{2}\) intercepts light with an intensity of \(10 \mathrm{~W} / \mathrm{m}^{2}\) from a camera strobe light. What radiation pressure is produced on the cardboard by the light?

Short Answer

Expert verified
The radiation pressure is approximately \( 3.33 \times 10^{-8} \text{ N/m}^2 \).

Step by step solution

01

- Convert Area to Square Meters

First, convert the area from square centimeters to square meters. Since 1 cm\textsuperscript{2} = 10\textsuperscript{-4} m\textsuperscript{2}, the area is given by:\[ A = 2.0 \text{ cm}^2 \times 10^{-4} \text{ m}^2/\text{cm}^2 = 2.0 \times 10^{-4} \text{ m}^2 \]
02

- Use the Radiation Pressure Formula

The radiation pressure for a totally absorbing surface is given by the formula:\[ P = \frac{I}{c} \]where \( I \) is the intensity and \( c \) is the speed of light ( \( c \approx 3 \times 10^8 \text{ m/s} \) ).
03

- Substitute Values

Substitute the given values into the formula:\[ I = 10 \text{ W/m}^2, \ c = 3 \times 10^8 \text{ m/s}\] So,\[ P = \frac{10 \text{ W/m}^2}{3 \times 10^8 \text{ m/s}} = \frac{10}{3 \times 10^8} \text{ N/m}^2 \]
04

- Simplify the Result

Simplify the result to find the radiation pressure:\[ P = \frac{10}{3 \times 10^8} \text{ N/m}^2 = \frac{1}{3 \times 10^7} \text{ N/m}^2 \approx 3.33 \times 10^{-8} \text{ N/m}^2 \]

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Key Concepts

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

Intensity of Light
The 'intensity of light' refers to the amount of energy the light wave carries per unit area, per unit time. In easier words, it tells us how strong or bright the light is.
For example, in our problem, the intensity is given as 10 W/m虏. This means that every square meter of the cardboard is hit with 10 watts of light energy from the strobe light.
The intensity of light is crucial in calculating radiation pressure, as radiation pressure is directly proportional to the intensity. Simply put, stronger light creates more pressure.
This is why we use the formula for radiation pressure: \( P = \frac{I}{c} \), where 'I' is the intensity of the light, and 'c' is the speed of light.
Always remember, the intensity value helps us understand how much energy is hitting the object.
Absorbing Surface
An 'absorbing surface' is a surface that takes in all the light energy falling on it without reflecting any light away.
In our problem, the cardboard is described as 'black and totally absorbing.' This means every bit of light energy is absorbed, not bounced back. Knowing if a surface is absorbing or reflecting is crucial, because it affects the calculation of radiation pressure.
For an absorbing surface, the radiation pressure formula simplifies to \( P = \frac{I}{c} \).
This tells us how much force per unit area the light is exerting on the surface.
Remember, if the surface were reflecting, the formula would be different. So, identifying the type of surface can change how we solve the problem.
Speed of Light
The 'speed of light' is one of the most important constants in physics. It is the rate at which light travels in a vacuum and is approximately 3 x 10^8 meters per second.
In shorter terms, light is incredibly fast!
When we solve for radiation pressure, we need to know the speed of light because it appears in our formula.
In our calculation, we used the speed of light value '3 x 10^8 m/s' to divide the light intensity and get the radiation pressure.
The formula we used is \( P = \frac{I}{c} \), where 'c' is the speed of light. This helps us understand how the light's intensity and speed combine to exert pressure on the absorbing surface.
Always use the correct value for the speed of light to get accurate results in your calculations.

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Most popular questions from this chapter

A plane electromagnetic wave has a maximum electric field of \(3.20 \times 10^{-4} \mathrm{~V} / \mathrm{m}\). Find the maximum magnetic field.

Laser eye surgery is carried out by delivering highly intense bursts of energy using electromagnetic waves. A typical laser used in such surgery has a wavelength of \(190 \mathrm{~nm}\) (ultraviolet light) and produces bursts of light that last for \(1 \mathrm{~ms}\). The laser delivers an energy of \(0.5 \mathrm{~mJ}\) to a circular spot on the cornea with a diameter of \(1 \mathrm{~mm}\). (The light is well approximated by a plane wave for the short distance between the laser and the cornea.) (a) Assuming that the energy of a single pulse is delivered to a volume of the cornea about \(1 \mathrm{~mm}^{3}\), and assuming that the pulses are delivered so quickly that the energy deposited has no time to flow out of that volume, how many pulses are required to raise the temperature of that volume from \(20^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) ? (Assume that the cornea has a heat capacity similar to that of water.) (b) Estimate the maximum strength of the electric field in one of these pulses.

During a test, a NATO surveillance radar system, \(\quad 20 .\) operating at \(12 \mathrm{GHz}\) and \(180 \mathrm{~kW}\) of power, attempts to detect an incoming stealth aircraft at \(90 \mathrm{~km}\). Assume that the radar beam is emitted uniformly over a hemisphere. (a) What is the intensity of the beam when it reaches the aircraft's location? The aircraft reflects radar waves as though it has a cross-sectional area of only \(0.22 \mathrm{~m}^{2}\). (b) What is the power of the aircraft's reflection? Assume that the beam is reflected uniformly over a hemisphere. Back at the radar site, what are the (c) intensity, (d) maximum value of the electric field vector, and (e) rms value of the magnetic field of the reflected (and now detected) radar beam?

The magnetic component of an electromagnetic wave in vacuum has an amplitude of \(85.8 \mathrm{nT}\) and an angular wave number of \(4.00 \mathrm{~m}^{-1}\). What are (a) the frequency of the wave, (b) the rms value of the electric component, and (c) the intensity of the light?

A horizontal beam of vertically polarized light of intensity \(43 \mathrm{~W} / \mathrm{m}^{2}\) is sent through two polarizing sheets. The polarizing axis of the first is at \(70^{\circ}\) to the vertical, and that of the second is horizontal. What is the intensity of the light transmitted by the pair of sheets?
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