I think this is quite instructive. First, imagine that one electron is held in place by an external force. As that electron does not move, the restraining force does no work. The second electron moves off to infinity with the calculation you have done. That electron, therefore, gets all the PE of the system. Once it's far enough away, you could release the first electron and there is no more PE to be squeezed out of the system.laser said:
But does this mean that dr is actually 2dr? Would this not double the potential energy?PeroK said:
If you label the electrons and take ##dx_1## as the differential along the path of one of the electrons, then ##dr = 2dx_1##. If you use that in your calculation, you get half the KE for the first electron. And the same KE for the other.laser said:
Coulomb's Law describes the electrostatic force between two charged particles. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as \( F = k_e \frac{|q_1 q_2|}{r^2} \), where \( F \) is the force, \( k_e \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.
Electric potential energy is the energy a charged particle possesses due to its position in an electric field. It is the work done to move a charge from a reference point (usually infinity) to a specific point in the field without any acceleration. The potential energy \( U \) between two point charges is given by \( U = k_e \frac{q_1 q_2}{r} \).
Potential energy in the context of Coulomb's Law is the energy associated with the electrostatic force between two charges. While Coulomb's Law gives the force between two charges, the potential energy represents the work needed to bring the charges to a certain separation distance. The potential energy \( U \) is derived from the integral of the force over distance, leading to \( U = k_e \frac{q_1 q_2}{r} \).
The sign of the charges determines the nature of the force between them. If the charges are of the same sign (both positive or both negative), the force is repulsive. If the charges are of opposite signs, the force is attractive. The potential energy also reflects this; it is positive for like charges (indicating repulsion) and negative for unlike charges (indicating attraction).
The distance between charges has an inverse-square relationship with the force and an inverse relationship with the potential energy. As the distance \( r \) increases, the force \( F \) decreases according to \( F = k_e \frac{|q_1 q_2|}{r^2} \), and the potential energy \( U \) decreases according to \( U = k_e \frac{q_1 q_2}{r} \). This means that both the force and potential energy diminish as the charges move farther apart.