Question

Two particles, \(\mathrm{A}\) and \(\mathrm{B}\), having same \(e / \mathrm{m}\) ratio are projected towards silver nucleus, in different experiments, with the same speed. The distance of closest approach will be (a) same for both (b) greater for A (c) greater for \(\mathrm{B}\) (d) depends on speed

Short answer

The distance of closest approach will be the same for both particles A and B.

Step-by-step solution

Understanding the concept of distance of closest approach

The distance of closest approach is the minimum distance between a charged particle and the nucleus of an atom when the particle is deflected due to the electrostatic force of the nucleus. It is determined by the initial kinetic energy of the particle and the charge of the nucleus, assuming that the particle stops at the closest distance before being repelled.

Recognizing the constant parameters

Since both particles A and B have the same charge-to-mass ratio (e/m) and are projected with the same speed, they have the same kinetic energy when approaching the silver nucleus. The charge of the silver nucleus is also unchanged in different experiments.

Relating kinetic energy to the distance of closest approach

The kinetic energy of the particles is converted into potential energy at the distance of closest approach. Given the kinetic energy and the charge of the nucleus are the same in both cases, the conversion from kinetic to potential energy will be the same, resulting in equal distances of closest approach for both particles.

Conclusion

Since the charge-to-mass ratio (e/m) and speed for both particles are the same during their respective experiments and the charge of the silver nucleus is constant, the distance of closest approach will be the same for both particles A and B.

Key concepts

Electrostatic Force

Imagine two kids on a playground pushing a spring toy back and forth. The electrostatic force works somewhat similarly, but on an atomic scale. It’s essentially the force experienced between charged particles such as electrons and protons. When charged particles are in close proximity, they exert a force on each other: like charges repel and opposite charges attract.

This force can be calculated by Coulomb's law, which states that the electric force (\( F \)) between two point charges is proportional to the product of their charges (\( q_1 \) and \( q_2 \) ), and inversely proportional to the square of the distance (\( r \) ) between their centers: \[ F = k \frac{|q_1 q_2|}{r^2} \]
where \( k \) is Coulomb's constant. In the context of our problem, a particle propelled towards a nucleus is influenced by the electrostatic force. This force decelerates the particle until its kinetic energy is converted into electrostatic potential energy at the distance of closest approach, which is the point where the particle momentarily comes to rest before being repelled.

Kinetic to Potential Energy Conversion

Have you ever rolled a ball up a hill? The faster it rolls, the higher it climbs before stopping and rolling back down. This can be likened to kinetic to potential energy conversion. Kinetic energy is the energy of movement, and in our scenario, it's the particle moving towards the nucleus. As it gets closer, it starts feeling the repulsive electrostatic force.

This is where the conversion happens: the particle's kinetic energy (motion) is transformed into electrostatic potential energy (stored energy from the repulsive force) until the kinetic energy is completely consumed. The point at which the particle's kinetic energy is fully converted to potential energy is the distance of closest approach. For our particles A and B, this process is the same as they have identical charges, masses, and speeds — leading to the same conversion rate and resulting in an equal distance of closest approach.

Charge-to-Mass Ratio (e/m)

The charge-to-mass ratio (e/m) is like the balance between the weight of your luggage and the limit you’re allowed on an airplane. It’s a measure that tells us how much charge a particle carries for a certain amount of mass. This ratio is significant because it affects how the particle behaves when moving through electric or magnetic fields.

In our problem, both particles A and B have the same charge-to-mass ratio and are given the same initial speed. This suggests that they respond equally to the electrostatic force exerted by the silver nucleus. Since this ratio for the two particles is identical, their acceleration, or deceleration in this case due to the electrostatic force, will also be the same. Therefore, their ability to penetrate the electrostatic shield of the nucleus before being repelled is identical, solidifying the conclusion that the distance of closest approach is the same for both particles.