Radiation Pressure: Videos & Practice Problems

Radiation pressure arises from the momentum of electromagnetic waves, which can exert force on objects upon absorption or reflection. When light is absorbed, the force is given by $f=\frac{I}{c}$, while for reflected light, it is $f=\frac{2I}{c}$. The pressure equations follow similarly, with absorbed pressure as $p=\frac{I}{c}$ and reflected pressure as $p=\frac{2I}{c}$. Understanding these concepts is crucial for applications in space technology and optics.

Radiation Pressure

Radiation pressure is a fascinating concept that plays a crucial role in various space technologies and applications. It arises from the interaction of electromagnetic waves, which not only carry energy but also possess momentum, despite having no mass. This momentum transfer occurs when light strikes an object, exerting a force that can cause the object to move. Understanding this phenomenon is essential for solving problems related to radiation pressure.

When light is absorbed by an object, it transfers its momentum, resulting in a force that accelerates the object. This scenario is akin to a completely inelastic collision, where two objects stick together and move as one. Conversely, when light is reflected, the momentum change is greater, leading to a larger force exerted on the object. This situation resembles an elastic collision, where the objects rebound off each other. Consequently, the force exerted by reflected light is always greater than that from absorbed light, which can be summarized as:

F_reflected = 2 × F_absorbed

The equations governing these forces are straightforward. For absorbed light, the force can be calculated using the formula:

F_absorbed = I × A / c

For reflected light, the formula is:

F_reflected = 2 × I × A / c

Here, I represents the intensity of the light, A is the area over which the light is distributed, and c is the speed of light (approximately 3 × 10⁸ m/s). The relationship between pressure, force, and area is given by:

P = F / A

From this, the radiation pressure for absorbed light can be expressed as:

P_absorbed = I / c

And for reflected light:

P_reflected = 2I / c

To illustrate these concepts, consider a practical example involving a laser pointer with an average power output of 5 milliwatts focused on a small area of the palm. If the hand completely absorbs the incoming light, the radiation pressure can be calculated using the intensity, which is derived from power divided by area:

I = P / A

Thus, the absorbed pressure becomes:

P_absorbed = (P / A) / c

Substituting the values, if the area is 1 × 10^-6 m², the resulting pressure is approximately 1.67 × 10^-5 pascals, indicating a very small pressure. To find the force exerted on the hand, one can use the relationship between pressure and force:

F_absorbed = P_absorbed × A

Calculating this yields a force of about 1.67 × 10^-11 newtons, which is indeed a minuscule force, consistent with the gentle nature of laser light on the skin.

In summary, radiation pressure is a critical concept in understanding how light interacts with matter, influencing various applications in technology and physics. The ability to calculate forces and pressures associated with light absorption and reflection is essential for practical problem-solving in this field.

A radio transmits a wave with intensity 27.0 W/m² towards a flat surface (perfectly reflecting) with area 2m². Calculate the force and radiation pressure on the surface.

• A. Pressure = $1.8\times10^{-7}Pa$ ; Force = $3.6\times10^{-7}N$
• B. Pressure = $9.0\times10^{-8}Pa$ ; Force = $1.8\times10^{-7}N$
• C. Pressure = $1.8\times10^{-7}Pa$ ; Force = $9.0\times10^{-8}N$
• D. Pressure = $9.0\times10^{-8}Pa$ ; Force = $3.6\times10^{-7}N$

Answer: (A)

Example 1

In this scenario, we explore the concept of using a laser beam to levitate an object, specifically a horizontal disc. The disc has a radius of 100 millimeters and a mass of 7.8 grams, while the laser beam, which exerts radiation pressure, has a radius of 50 millimeters and is positioned 2.75 meters away from the disc. The goal is to determine the power required for the laser to balance the forces acting on the disc, allowing it to float.

To achieve levitation, the upward force generated by the radiation pressure from the laser must equal the downward force of gravity acting on the disc. This relationship can be expressed mathematically as:

$$ F_{\text{radiation}} = F_g $$

Where \( F_g \) (the force of gravity) is calculated using the formula:

$$ F_g = m_{\text{disc}} \cdot g $$

Here, \( m_{\text{disc}} \) is the mass of the disc (converted to kilograms: 0.0078 kg) and \( g \) is the acceleration due to gravity (approximately 9.8 m/s²).

For the radiation pressure, since the disc completely absorbs the light, we use the formula for the force due to absorbed radiation:

$$ F_{\text{absorbed}} = \frac{I \cdot A}{c} $$

Where \( I \) is the intensity of the laser, \( A \) is the area of the laser beam, and \( c \) is the speed of light (approximately \( 3 \times 10^8 \) m/s). The intensity \( I \) can be expressed in terms of power \( P \) and area \( A \) as:

$$ I = \frac{P}{A} $$

Substituting this into the radiation pressure equation gives:

$$ F_{\text{absorbed}} = \frac{P}{A} \cdot A \cdot \frac{1}{c} = \frac{P}{c} $$

Setting the two forces equal to each other, we have:

$$ \frac{P}{c} = m_{\text{disc}} \cdot g $$

Rearranging this equation allows us to solve for the power \( P \) of the laser:

$$ P = m_{\text{disc}} \cdot g \cdot c $$

Substituting the known values:

$$ P = 0.0078 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 3 \times 10^8 \, \text{m/s} $$

Calculating this yields:

$$ P \approx 2.29 \times 10^7 \, \text{W} $$

This result is equivalent to approximately 222.9 megawatts, indicating that an immense amount of power is required to achieve levitation through radiation pressure. This example illustrates the significant forces involved in radiation pressure and the challenges associated with manipulating such forces in practical applications.

Example 2

In this scenario, we explore the concept of using a spacecraft equipped with reflective sails to harness sunlight for propulsion, a promising technology for low-cost space travel. The spacecraft, a 400-kilogram satellite, utilizes its reflective sails to capture sunlight, generating a force known as radiation pressure. This force propels the satellite forward, allowing it to gain speed without relying on traditional fuel sources.

The total area of the reflective sails is crucial for calculating the force exerted on the satellite. Each sail has an area of 5,000 square meters, and with two sails, the total area becomes 10,000 square meters. The intensity of sunlight at the satellite's location is approximately 1350 W/m². For reflective surfaces, the force due to radiation pressure can be calculated using the formula:

F_{\text{reflected}} = \frac{2 \cdot I \cdot A}{c}

Here, I represents the intensity of sunlight, A is the total area of the sails, and c is the speed of light, approximately \(3 \times 10^8\) m/s. Plugging in the values:

F_{\text{reflected}} = \frac{2 \cdot 1350 \, \text{W/m}^2 \cdot 10,000 \, \text{m}^2}{3 \times 10^8 \, \text{m/s}} \approx 0.09 \, \text{N}

This force, while small, is constant and will continue to accelerate the satellite over time. To determine the satellite's velocity after one year, we apply kinematic equations. Assuming the satellite starts from rest, the initial velocity (\(v_0\)) is 0. The acceleration (\(a\)) can be calculated using Newton's second law:

a = \frac{F}{m} = \frac{0.09 \, \text{N}}{400 \, \text{kg}} = 2.25 \times 10^{-4} \, \text{m/s}^2

Next, we convert one year into seconds for our calculations:

t = 1 \, \text{year} = 365 \, \text{days} \times 24 \, \text{hours/day} \times 60 \, \text{minutes/hour} \times 60 \, \text{seconds/minute} \approx 31,536,000 \, \text{seconds}

Now, we can find the final velocity (\(v_f\)) using the formula:

v_f = v_0 + a \cdot t = 0 + (2.25 \times 10^{-4} \, \text{m/s}^2) \cdot (31,536,000 \, \text{s}) \approx 7,100 \, \text{m/s}

This calculation illustrates that even a small force can lead to significant changes in velocity over extended periods, demonstrating the potential of solar sails for long-duration space missions. The satellite, propelled by sunlight, could achieve impressive speeds as it travels through the solar system.

Here’s what students ask on this topic:

What is radiation pressure and how is it calculated?

Radiation pressure is the pressure exerted by electromagnetic waves on a surface due to the transfer of momentum. It can be calculated for two scenarios: absorption and reflection. For absorbed light, the force is given by $f=\frac{I}{c}$, where $I$ is the intensity of the light and $c$ is the speed of light. For reflected light, the force is $f=\frac{2I}{c}$. The corresponding pressure equations are $p=\frac{I}{c}$ for absorbed light and $p=\frac{2I}{c}$ for reflected light.

How does radiation pressure affect space technology?

Radiation pressure plays a significant role in space technology, particularly in the design and operation of solar sails. Solar sails use the momentum of photons from the Sun to propel spacecraft without the need for traditional fuel. The force exerted by radiation pressure can be harnessed to accelerate, decelerate, or change the direction of a spacecraft. This technology offers a sustainable and efficient method for long-duration space missions, reducing the dependency on chemical propulsion systems.

Why does reflected light exert more force than absorbed light?

Reflected light exerts more force than absorbed light because the change in momentum is greater. When light is absorbed, it transfers its momentum to the object, resulting in a force given by $f=\frac{I}{c}$. However, when light is reflected, it reverses direction, effectively doubling the change in momentum. This results in a force of $f=\frac{2I}{c}$, making the force exerted by reflected light twice as large as that of absorbed light.

What are the equations for radiation pressure in terms of intensity and speed of light?

The equations for radiation pressure depend on whether the light is absorbed or reflected. For absorbed light, the pressure is given by $p=\frac{I}{c}$, where $I$ is the intensity of the light and $c$ is the speed of light. For reflected light, the pressure is $p=\frac{2I}{c}$. These equations highlight the relationship between the intensity of the electromagnetic waves and the resulting pressure exerted on a surface.

How do you calculate the force exerted by radiation pressure on an object?

To calculate the force exerted by radiation pressure on an object, you need to know the intensity of the light and the area over which it is distributed. For absorbed light, the force is given by $f=\frac{IA}{c}$, where $I$ is the intensity, $A$ is the area, and $c$ is the speed of light. For reflected light, the force is $f=\frac{2IA}{c}$. These equations allow you to determine the force based on the light's intensity and the surface area it impacts.