Question

A particle moving in a conservative force field from point $A$ to point $B$. ${U_A}$ and ${U_B}$ are the potential energies of the particle at the point $A$ and $B$ and ${W_C}$ is the work done by conservative forces in the process of taking the particle from $A$ and $B$:
(A) ${W_C} = {U_B} - {U_A}$
(B) ${W_C} = {U_A} - {U_B}$
(C) ${W_C} = {U_A} + {U_B}$
(D) ${W_C} = {U_B} + {U_A}$

Answer 1

Hint
When the particle is in the conservative force field from one point to another point, then the potential energies of the particles from one point to another point is given by the work done. The work done by the conservative forces in the process of taking the particle from one point to another point is the difference of the potential differences.

Complete step by step answer
The conservative forces will occur when the work done by the force on the object is independent to the path of the object. The work done by the conservative force will depend only on the end point of the motion of the force. The most suitable example for the conservative force is gravity. The potential energy and the work done are likely the same things, like the kinetic energy and the work done are the same thing. The potential energy is the state that the system stores the energy of the motion, while the work done is the way of changing the energy from one object to another object.
When the changes in the potential energy of the moving object, which does not depend on the path of the motion, but it depends only on initial and final positions of the displacement of the moving object. Then the force field is called conservative force fields. Then, the work done is given by the difference of the potential energy, then
$\Rightarrow {W_C} = {U_A} - {U_B}$
Hence, the option (B) is the correct answer.

Note
The potential is equal to the amount of work done by the object on the position of the object. But here the object is in two positions, initial position and final position, so the work done is given by the difference of the potentials at the two points.