Question

Charged particle X moves directly towards another charged particle Y. For the X plus Y system, the total momentum is p and the total energy is E. This question has multiple correct options
A) p and E is conserved if both X and Y are free to move
B) is true only if X and Y have similar charges.
C) If Y is fixed, E is conserved but not p.
D) If Y is fixed, neither E nor P is conserved

Answer 1

Hint: All systems, whether like or unlike charges, conserve their total energy if no external energy is supplied or taken away. The total momentum of a system is conserved as long as there is no external force acting it.

Complete step by step answer
Let's first consider the case when the two charges are completely free. We know from the law of conservation of momentum that If no force is acting on a system and if Newton's third law is valid, the momentum of the system would be conserved. Here, the force between the two charges is an internal force and Force on X due to Y is the same as that on Y due to X. So the momentum of the whole system is conserved.
Also, here since there exist no energy losses, the total energy of the system remains the same. So option A, which says the energy and momentum of the system are conserved if X and Y are free is true.
Now let us analyse the case when charge Y is fixed. When this happens, we are exerting an external force to prevent the charge from moving. So this adds an external force to the system and hence, the momentum of the system is no longer conserved.
While considering energy conservation, The force that prevents Y from moving also has to be considered. If this external force does not do any work on the system, then the energy of the system should be conserved.
We know work is said to be done if A force F acting on a particle creates a displacement of x on it.
Here, the force acting on Y is just holding it there and creating no displacement. So it is doing zero work on the system and hence, the total energy of the system must be conserved.
This corresponds to option C and hence the two correct options are A and C.

Note: We should not misinterpret the conservation of energy as a conservation of kinetic energy. Collisions may lead to a loss in kinetic energy but the total energy is conserved.
It is true in general that an isolated system has constant energy. The charges need not be similar as in option B to apply energy conservation.