To understand the relationship and differences between electric potential and electric potential energy.
In this problem we will learn about the relationships between electric force \vec{F}, electric field \vec{E}, potential energy U, and electric potential V. To understand these concepts, we will first study a system with which you are already familiar: the uniform gravitational field.
First we review the force and potential energy of an object of mass m in a uniform gravitational field that points downward (in the -\hat{z} direction), like the gravitational field near the earth's surface.
Part A Find the force F_vec(z) on an object of mass m in the uniform gravitational field when it is at height z=0.
Express F_vec(z) in terms of m, z, z_unit, and g.
F_vec(z) = -mg\hat{z}
Because we are in a uniform field, the force does not depend on the object's location. Therefore, the variable z does not appear in the correct answer.
Part B Now find the gravitational potential energy U(z) of the object when it is at an arbitrary height z. Take zero potential to be at position z=0.Express U(z) in terms of m, z, and g. Note that because potential energy is a scalar, and not a vector, there will be no unit vector in the answer.
U(z) = mgz
Part C In what direction does the object accelerate when released with initial velocity upward?
1. upward
2. downward
3. upward or downward depending on its mass
4. downward only if the ratio of g to initial velocity is large enough
Now consider the analogous case of a particle with charge q placed in a uniform electric field of strength E, pointing downward (in the -\hat z direction)
Part D Find F_vec(z), the electric force on the charged particle at height z.