Electrical Potential & Work

Below is a diagram of a box on an inclined surface. The work done from point A to B is zero because the cos(90)=0. The work from point B to C is negative and equal to mghcos(180) for this case. The force is negative because the weight force is acting downward. 
Work in a gravitational field is path independent. Electric fields are conservative and able to release all energy.

When the path is perpendicular to the axis, there is no work done. A path parallel to the axis means that work is done. The path at an angle means some work is done. 

Here we used the voltage integral and the formula for work done per unit charge to find a formula for voltage in terms of charge and distance.

Equipotential difference can be calculated using the formula derived below.

Below is a diagram of 2 charges on a 3 dimensional plane. We chose 2 observation locations and used the given charges and separation distances to calculate the electrical potential at these locations.

The V-Python program was used to display and calculate the electrical potential at 2 observation points between 2 charged spheres. The programing is shown as well. 

The calculations below were performed for practice.

The diagram and calculations below were verified on V-Python.

Summary:

• Work in a gravitational field is path independent. 
• Electric fields are conservative and able to release all energy.
• When the path is perpendicular to the axis, there is no work done. 
• A path parallel to the axis means that work is done. The path at an angle means some work is done. 
• The electrical potential changes with different observation points. The values for the potential can be found through simple calculations or by using V-Python.