Problem 55 - Master Resource Book in JEE Main Physics

Question

Consider aiming a beam of free electrons towards free protons. When they scatter, an electron and a proton cannot combine to produce a H-atom. [NCERT Exemplar] (a) because of energy conservation (b) without simultaneously releasing energy in the form of radiation (c) because of momentum conservation (d) because of angular momentum conservation

Step-by-Step Solution

Answer

Electrons and protons cannot form hydrogen atoms without releasing energy as radiation (option b).

Step 1: Understanding the Problem

The problem is about understanding why an electron and a proton do not form a hydrogen atom when scattered towards each other. We need to consider principles of physics like energy, momentum, and angular momentum conservation, and if radiation plays a role.

Step 1: Consider Energy Conservation

For a hydrogen atom to form, the system must lose energy to stabilize at a lower state. If no mechanism (like photon emission) is available, the electron and proton cannot lose energy to form a bound state. This option relates to the requirement for radiation emission.

Step 2: Examine Radiation Requirement

In forming a hydrogen atom, energy would be released as a photon due to the electron transitioning to a lower energy state. Without this release of energy in radiation form, the electron remains unbound, or free.

Step 3: Assess Momentum Conservation

Momentum conservation involves the total momentum before and after collision remaining constant. While important in collisions, this alone does not prevent electrons and protons from binding if energy can be released.

Step 4: Evaluate Angular Momentum Conservation

Angular momentum conservation also plays a role in atomic systems. However, its conservation does not inherently prevent electron-proton binding without energy release.

Step 6: Conclusion: Identify the Correct Reason

Among all considerations, the absence of a radiation mechanism (option b) specifically addresses why they cannot combine because it prevents energy loss needed for a bound state.

Key Concepts

Energy Conservation

In the context of electron-proton scattering, understanding energy conservation is crucial. Energy conservation states that the total energy of an isolated system remains constant over time. When you aim a beam of electrons at protons with the hope of forming a hydrogen atom, there are energy changes to consider. However, simply colliding an electron and a proton is not enough to create a hydrogen atom.

For a hydrogen atom to form, the system must lose energy to reach a more stable, lower energy state. This loss of energy often happens through the emission of a photon, which is a tiny packet of light. Without this mechanism, the total energy remains too high for the electron and proton to form a stable bond. Thus, the conservation of energy plays a direct role in preventing the formation of a hydrogen atom without this energy loss.

Momentum Conservation

Momentum conservation is another important concept in understanding why a simple collision between an electron and a proton does not result in the formation of a hydrogen atom. According to the law of momentum conservation, the total momentum of a system must remain constant before and after the collision.

During scattering, both the electron and proton have certain velocities, and their combined momentum must equal the momentum after interaction. However, even if momentum is conserved, it does not inherently allow for the formation of a bound state such as a hydrogen atom. For binding, energy must be released, refocusing our attention back to the need for a mechanism to emit energy as radiation, which is not directly addressed by momentum conservation alone.

Radiation Emission

Radiation emission is a key part of understanding why an electron and a proton do not bind to form a hydrogen atom during a collision. When an electron transitions from a high energy state to a lower one, it often releases energy in the form of radiation, specifically photons.

In the absence of radiation emission, the electron cannot lose enough energy to become bound to the proton. This means they cannot stabilize at the lower energy state required for forming a hydrogen atom. Therefore, without energy loss through radiation, the electron remains unbound, maintaining its free nature. Thus, radiation emission is fundamental to the process of electron-proton binding.

Angular Momentum Conservation

Angular momentum conservation is an important concept in physics, particularly in systems involving rotational dynamics. It states that the angular momentum of a system remains constant in the absence of external torques. In the scattering of electrons and protons, angular momentum is indeed conserved.

However, similar to momentum conservation, the conservation of angular momentum alone does not explain the barrier to forming a hydrogen atom. Angular momentum must indeed be considered in quantum systems where electron orbitals have defined angular momentum values. Yet, without the simultaneous emission of energy, the conservation of angular momentum does not enable binding. It primarily ensures that any formed structure maintains its rotational properties, rather than influencing the formation of the structure itself.