Question

A radiation of energy \(\mathrm{E}\) falls normally on a Perfect reflecting surface. The momentum transferred to the surface is. (A) \((\mathrm{E} / \mathrm{c})\) (B) \((2 \mathrm{E} / \mathrm{c})\) (C) \(\left(\mathrm{E} / \mathrm{c}^{2}\right)\) (D) Ec

Short answer

The momentum transferred to the reflecting surface is (B) \((2 \mathrm{E} / \mathrm{c})\).

Step-by-step solution

Understand the concept of reflection and conservation of momentum.

During reflection, the direction of the momentum changes, but the magnitude remains the same. As the surface is perfectly reflecting, the momentum before and after reflection will be the same in magnitude but opposite in direction. Therefore, the total momentum transferred to the reflecting surface will be twice the initial momentum.

Calculate the initial momentum of the radiation.

Using the given relationship E = pc, we can find the initial momentum of the radiation. To find the value, we need to rearrange the equation as p = E/c.

Calculate the momentum transferred to the surface.

As mentioned earlier, since the surface is perfectly reflecting, all the initial momentum of the radiation is transferred to the surface. Therefore, the momentum transferred to the surface is 2*(initial momentum). From step 2, we found the initial momentum to be E/c, so the momentum transferred to the surface will be 2*(E/c) = 2E/c. The correct answer is (B) \((2 \mathrm{E} / \mathrm{c})\).

Key concepts

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics. It states that within a closed system, the total momentum remains constant if no external forces are acting upon it. This concept is crucial in understanding interactions like collisions and reflections. When light reflects off a surface, like in the case of a perfectly reflecting surface, the conservation of momentum requires that the momentum lost by the photons is gained by the surface. The momentum is transferred without any losses, assuming no other forces are at play. Hence, for systems involving light or particles reflecting off surfaces, conservation of momentum helps us predict the resulting motion accurately.

Perfect Reflection

Perfect reflection refers to the scenario where a wave, such as light or radiation, bounces off a surface without any loss of energy. In physics, this assumes that all the incident energy is reflected, and none is absorbed by the surface. This concept is idealized and mostly applicable in theoretical discussions or under very controlled conditions.
In perfect reflection, not only is energy conserved, but the direction of momentum also changes. The incident photon momentum reverses after reflection, doubling the overall momentum transfer to a perfectly reflecting surface. This is a key factor in calculating how much momentum a surface receives when struck by radiation, leading directly to the conclusion that the transferred momentum equals twice the initial momentum of the radiation.

Radiation Pressure

Radiation pressure is the pressure exerted upon surfaces due to electromagnetic radiation. This phenomenon arises because photons, despite being massless, carry momentum. When these photons strike a surface, they transfer momentum, causing a small force to be exerted on that surface. This effect is utilized in mechanisms such as solar sails in space, where radiation pressure provides propulsion.

• Radiation pressure is typically very small, but it becomes significant in spaces where other forces are minimal.
• It is calculated using the momentum transfer rate; in perfect reflections, this transfer rate is maximized due to the doubling of momentum.

Understanding radiation pressure helps in applications ranging from experimental physics to astrodynamics, showing how even massless particles can influence motion.

Photon Momentum

Despite being devoid of mass, photons possess momentum given by the relationship:\[ p = \frac{E}{c} \]where \( p \) is the momentum, \( E \) is the energy of the photon, and \( c \) is the speed of light. This relationship forms the basis of calculating momentum transfer in light interactions with surfaces, such as reflections.
The concept of photon momentum is crucial in topics like quantum mechanics and electromagnetism, as it bridges classical and modern physics. By understanding and applying the properties of photon momentum, we can predict how radiation influences materials it interacts with, such as creating forces or causing pressure on reflecting surfaces.