Animated Solution for Physics - Electromagnetic Waves: A radiation of energy $E$ falls normally on a perfectly reflecting surface. The momentum transferred to the surface is

Welcome to a fascinating exploration of the dual nature of light and electromagnetic waves! In this problem, we are going to dive deep into the concept of radiation pressure and momentum transfer. It might seem counterintuitive at first—how can light, which has no rest mass, carry momentum? But as we will see, this is one of the most profound discoveries of modern physics.

When we think of momentum, we usually picture a heavy object moving at a certain speed, like a bowling ball or a speeding car. The classical formula for momentum is simply mass times velocity ($p=mv$). However, light is made of photons, which are massless particles. So, how can they have momentum?

The answer lies in the realm of relativity and electromagnetism. According to the theory of electromagnetism and Einstein's energy-momentum relation, the energy $E$ of a photon is directly proportional to its momentum $p$. The relationship is beautifully simple:

where $c$ is the speed of light in a vacuum. Rearranging this, we find that the momentum of a photon (or a beam of radiation) is given by:

This equation is our golden key. It tells us that any radiation carrying energy $E$ also carries a momentum equal to $E/c$.

The Incident Beam

Imagine you are standing in a laboratory, and you shine a laser beam directly onto a mirror. The problem states that a radiation of energy $E$ falls normally on a perfectly reflecting surface. The word 'normally' is crucial here. It means the radiation strikes the surface at exactly a 90-degree angle, perpendicular to the plane of the surface. This simplifies our problem into a one-dimensional scenario, allowing us to focus entirely on the magnitude and sign of the momentum without worrying about complex angles or vector components.

Let's define our coordinate system. Let the direction towards the surface be the positive direction. The incident radiation is moving towards the surface, so its initial momentum $p_i$ is positive. Using our golden key, we can write:

A Reversal of Fate

Now, what happens when the radiation hits the surface? The problem specifies that the surface is perfectly reflecting. This is an idealization, meaning that 100% of the incident energy is bounced back. None of it is absorbed or transmitted.

Because the energy $E$ remains unchanged, the magnitude of the momentum must also remain unchanged. However, the radiation is now moving away from the surface. In our coordinate system, this is the negative direction. Therefore, the final momentum $p_f$ of the reflected radiation is:

This negative sign is not just a mathematical formality; it represents a complete reversal of the physical direction of the light beam.

Change in Momentum

To find out how much momentum the surface gained, we first need to determine how much momentum the radiation lost. We do this by calculating the change in momentum of the radiation, denoted by $\Delta p$.

The change in any quantity is always its final value minus its initial value:

Let's substitute the values we found:

The radiation experiences a change in momentum of $-2E/c$. The negative sign indicates that the impulse acting on the radiation is directed away from the surface, which makes perfect sense—the surface pushed the light away!

Momentum Transferred

Now, we invoke one of the most fundamental laws of nature: Newton's Third Law of Motion. For every action, there is an equal and opposite reaction.

When the radiation hits the surface, it exerts a force on it, and the surface exerts an equal and opposite force back on the radiation. Consequently, the momentum transferred to the surface (let's call it $\Delta p^{\prime }$) must be equal and opposite to the change in momentum of the radiation.

Substituting our result for $\Delta p$:

And there we have it! The surface absorbs a momentum of $2E/c$ in the positive direction (the original direction of the light).

This result is incredibly important in physics. It is the underlying principle behind solar sails, a proposed method of spacecraft propulsion that relies entirely on the momentum transferred by sunlight reflecting off large, mirror-like sails. By understanding this simple problem, you are grasping the mechanics that could one day take humanity to the stars!

A Thought Experiment

To truly master this concept, it is always helpful to ask 'what if?' What if the problem had stated that the surface was perfectly absorbing instead of perfectly reflecting?

Imagine the surface is painted pitch black. When the radiation strikes it, all the energy $E$ is absorbed, and the light ceases to exist. In this scenario, the initial momentum is still $p_i=E/c$. However, because the light is absorbed, there is no reflected beam. The final momentum of the radiation is simply zero:

Now, let's calculate the change in momentum for this absorbing case:

Applying Newton's Third Law again, the momentum transferred to the surface would be:

Notice the profound difference! A perfectly reflecting surface receives twice as much momentum as a perfectly absorbing surface. This is because the reflecting surface not only has to stop the incoming light but also has to 'throw' it back in the opposite direction, requiring a much larger impulse.

This is a classic trap in competitive exams like JEE and NEET. Examiners love to swap the words 'reflecting' and 'absorbing' to test if you are truly visualizing the physics or just memorizing formulas. Always read the problem statement carefully and visualize the physical process before writing down any equations.

We have successfully navigated through the problem. By understanding the relationship between energy and momentum for electromagnetic waves, carefully defining our coordinate system, and applying the universal principle of momentum conservation, we arrived at the correct answer. The momentum transferred to a perfectly reflecting surface by radiation of energy $E$ is indeed $2E/c$. Keep practicing, keep visualizing, and the beautiful logic of physics will always guide you to the right solution!