An approach to conservation of electrical mechanical energy

[ehabmozart]

Homework Statement

A point charge q1=5μC is held fixed in space. From a horizontal distance of 6.00 cm, a small sphere with mass 4g and charge 2μC is fired toward the fixed charge with an initial speed of 40.0 m/s. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is 25.0 m/s?

Homework Equations

Conservation of Energy and Coulomb's law

The Attempt at a Solution

I am able to solve the full question but I am wondering is it possible to use conservation of energy in this case. I mean the work has been done by an external force, so how is it possible to say
Ka+Ua = Kb+Ub ALONE... Shouldn't we add W other to the initial mechanical energy? If someone can explain with analogy to gravity, it would be great too :)

 

[Sunil Simha]

I am able to solve the full question but I am wondering is it possible to use conservation of energy in this case. I mean the work has been done by an external force, so how is it possible to say
Ka+Ua = Kb+Ub ALONE... Shouldn't we add W other to the initial mechanical energy? If someone can explain with analogy to gravity, it would be great too :)

What is the work done here (what is W)? Who/what is the external agent?

 

[ehabmozart]

What is the work done here (what is W)? Who/what is the external agent?

The external agent is the force given by the firing.. It is not a conservative force?

 

[Sunil Simha]

The external agent is the force given by the firing.. It is not a conservative force?

Say the 2μC sphere is a bullet, the gun and the bullet are at rest right? So all the work that the guy holding the gun to bring the bullet up to the point of firing has simply gone into increasing the potential energy of the bullet (while bringing it from infinity to that point). So W is simply the potential energy at that point.

 

[Nickie]

I would say that conservation of energy can be applied in this case, but it is important to consider the different forms of energy involved. In this situation, the initial mechanical energy of the system is solely due to the initial speed of the sphere. As the sphere approaches the fixed charge, its kinetic energy is converted into electrical potential energy due to the Coulombic attraction between the charges. Thus, at any given instant, the total energy (mechanical + electrical) of the system remains constant.

When the sphere's speed is 25.0 m/s, it has lost some of its initial kinetic energy, which has been converted into electrical potential energy. This means that the sum of the sphere's final kinetic energy and the electrical potential energy between the two charges must be equal to the initial mechanical energy of the system. Therefore, the equation Ka + Ua = Kb + Ub can be used to solve for the acceleration of the sphere at this instant.

To explain this using an analogy to gravity, we can imagine a ball rolling down a hill. The ball has initial kinetic energy due to its speed, but as it rolls down the hill, this kinetic energy is converted into potential energy due to gravity. At any given point on the hill, the total energy of the system (kinetic + potential) remains the same. Similarly, in the given problem, the initial kinetic energy of the sphere is converted into electrical potential energy, but the total energy of the system remains constant.