Example Problems involving Electric Fields and Potential Energy

G20-2. Example Problems involving Electric Fields and Potential Energy

When doing conservation of energy problems involving electric forces, the same methods apply as we've used in the past for gravitation and other forces. For review, these methods are given here. For electrical forces, one determines electrical potential energy using one of the two formulas given in Rows 8 and 9 of the table on this page. We give some example problem solutions below.

Example Problem 1: View this animation to see the problem situation. We'll solve the problem using two different methods.

Given: m = mass of proton = 1.67E-27 kg
           e = charge of proton = +1.60E-19 C
           E = electric field = 1250 N/C in the +x direction
           x₀ = initial position of proton = 0.100 m
           x = position of proton at turn-around point = -0.0345 m

Goal: Find the initial velocity, v₀, of the proton.

Method 1. Conservation of Energy

System: proton
External forces: none
Initial state: At t = 0, the proton has velocity v₀.
Final state: At t = 1.5E-6 s, the proton has velocity 0.

Notes:

We substituted Δs = -Δx. That's because Δs represents the displacement in the direction of the field. Since the proton is displaced opposite the field, Δs is the negative of Δx. Had we not made that substitution, we would have ended up with the square root of a negative number.
We selected the negative root, since we know the initial velocity is to the left.

Method 2. Net Force

Notes:

There's only one force acting on the proton as shown in the force diagram.
Since the field is uniform, the acceleration is also uniform. Therefore, we use a dvat to complete the solution.
The result is same as with Method 1.

Example Problem 2: Now we'll introduce an external force. View this animation to see the problem situation. Once again, we'll solve the problem using two different methods.

Goal: Find the initial velocity, v₀, of the proton.

Method 1. Conservation of Energy

System: proton
External forces: F_ext
Initial state: At t = 0, the proton has velocity v₀.
Final state: At t = 1.55E-6 s, the proton has velocity 0.

Note that in the calculation of work done by the external force, we use the absolute value of the displacement. Since the angle between the force and the displacement is 0°, this will give a positive value for work, as expected. Later, we substitute (x_o - x) for |Δx|.

Method 2. Net Force

There are two forces acting on the proton as shown in the force diagram.

A dvat is used to complete the solution.

The same result is obtained as for the conservation of energy method.