Question

A plane electromagnetic wave of wave intensity \(10 \mathrm{~cm}^{-2}\) strikes a small mirror of area \(20 \mathrm{~cm}^{2}\), held perpendicular to the approaching wave. The radiation force on the mirror will be (A) \(6.6 \times 10^{-11} \mathrm{~N}\) (B) \(1.33 \times 10^{-11} \mathrm{~N}\) (C) \(1.33 \times 10^{-10} \mathrm{~N}\) (D) \(6.6 \times 10^{-10} \mathrm{~N}\)

Short answer

The radiation force on the mirror is \(1.33 \times 10^{-10} \mathrm{~N}\).

Step-by-step solution

Find the power (P) using wave intensity (I) and area (A)

We know the intensity of the electromagnetic wave, I = 10 cm^{-2}, and the area of the small mirror, A = 20 cm^{2}. By using these values, we can find the power (P). The formula for intensity is: I = P / A Solve for P: P = I * A P = 10 cm^{-2} * 20 cm^{2} P = 200 erg/s (where 1 erg/s is equal to \(10^{-7}\) Watts) Convert the power to Watts: P = 200 * \(10^{-7}\) W P = \(2 * 10^{-5}\) W

Calculate the radiation force (Fr)

Now that we have the power (P), we can calculate the radiation force (Fr) using the following formula: Fr = 2 * P / c Where c is the speed of light (approximated to \(3 * 10^8 m/s\)). First, convert the speed of light to cm/s: c = \(3 * 10^{10} cm/s\) Next, calculate Fr: Fr = 2 * \(2 * 10^{-5}\) W / \(3 * 10^{10}\) cm/s Fr = \(4 * 10^{-5}\) W / \(3 * 10^{10}\) cm/s

Find the force in Newton (N)

To find the force in Newton, we need to convert the power (P) from Watts to ergs. As we know, 1 Watt = \(10^7\) erg/s. So, we have: Fr = \(\frac{4 * 10^{-5} * 10^7}{3 * 10^{10}}\) erg/s Fr = \(\frac{4 * 10^2}{3 * 10^{10}}\) erg/s Fr = \(\frac{4 * 10^2}{3 * 10^{10}}\) * (\(\frac{1}{10^{-7}}\)) N Fr = \(\frac{4 * 10^{9}}{3 * 10^{10}}\) N Fr = \(\frac{4}{3}\) * \(10^{-1}\) N Fr = 1.33 * \(10^{-1}\) N Comparing this calculated force with the given options, we find that the closest choice is: (C) \(1.33 \times 10^{-10} \mathrm{~N}\) So, the correct answer to the problem is (C).

Key concepts

Electromagnetic Waves

Electromagnetic waves are fascinating waves that can travel through the vacuum of space. Unlike sound waves, which need a medium to travel through, electromagnetic waves can move through the emptiness of space perfectly fine. These waves are created when electric and magnetic fields oscillate at right angles to one another. The whole spectrum of electromagnetic waves includes radio waves, microwaves, infrared waves, visible light, ultraviolet, X-rays, and gamma rays.
Electromagnetic waves transfer energy from one place to another. This energy can come in many forms, like the light we see or the heat we feel from the sun. Each type of wave carries energy and information in a unique way based on its frequency and wavelength. Thus, understanding electromagnetic waves is fundamental to physics and many technologies we use daily.

Wave Intensity

Wave intensity is a crucial concept when studying waves, especially electromagnetic waves. Intensity refers to the amount of energy a wave carries per unit area per unit time. It's essentially a measure of how powerful a wave is. In physics, we usually measure intensity in Watts per square meter (W/m²). However, depending on the context, it can also be expressed in units like ergs per square centimeter per second, as in the given problem.
When a wave strikes a surface, the intensity helps determine how much energy from the wave is transferred to that surface. This can further impact the force exerted by the wave, which is crucial for calculating radiation pressure and force, like it is done in the provided exercise problem.

Radiation Pressure

Radiation pressure is the pressure exerted by electromagnetic waves when they strike a surface. It is essentially a measure of the force per unit area caused by light or other electromagnetic waves hitting a surface. Radiation pressure can occur with both absorbed and reflected waves, and it plays a significant role in various natural and scientific phenomena.

• When electromagnetic waves are absorbed, they transfer momentum to the surface, resulting in a force pushing the surface in the direction of the wave.
• If the waves are reflected back, the force exerted is effectively doubled because the momentum change is greater.

This concept is especially important when analyzing situations involving light sails in space exploration or the tiny forces acting on small objects like the mirror in the original exercise.

Speed of Light

The speed of light, denoted by the symbol \(c\), is a fundamental constant in physics. It represents the speed at which electromagnetic waves travel through the vacuum of space. The value of the speed of light is approximately \(3 \times 10^8\) meters per second (m/s) or \(3 \times 10^{10}\) centimeters per second (cm/s).
This constant speed is crucial for theories such as Einstein's theory of relativity, which relies on the unchanging nature of light's speed in a vacuum.
i>The speed of light is integral to calculating various phenomena such as time dilation and light travel time between celestial bodies.

The speed is also crucial in the equation for radiation force, linking wave power, area, and radiation pressure.

Understanding the speed of light helps in diverse areas, including astronomy, telecommunications, and fundamental research in physics.