Electrical Potential & Equipotential Surfaces

Here we derived a formula for the electrical potential at a distance R at point P. We then used this formula to solve a given problem. 

The next derivation is for the electrical potential at a distance R at a point P. The distance R is equal to x in this case. 

Below is the electrical potential in terms of theta.

This derivation is for the electrostatic force where r is found using trigonometry. There is no y-component in this derivation.

Here we solved a problem for the electrical potential due to a finite length line charge.

Equipotential Surfaces allow for charges to travel without doing any work. The potential has the same value at all points. No energy is required to move on the same equipotential line; therefore, work is done.

Below is an image of a proton and an electron. The field lines are leaving the proton and entering the electron. 

The conductive paper below was connected to a board. We used a multimeter to measure the voltages at different distances along the conductive paper. We took readings from the negative and the positive x-axis.

For gravity, the potential energy of each kg of mass decreases as the mass travels in the direction of the gravitational field, and increases as it travels opposite the direction of the gravitational field. A positive charge will go from higher to lower potential energy along the direction of the electric field. 

We then found the change in voltage per centimeter and meter, and created a table of our findings.

Next, we found the work done when a charge is moved certain distances. The calculations were as follows:

This graph depicts the relationship between the electrical potential and distance from our results.

Summary:

• Equipotential Surfaces allow for charges to travel without doing any work. 
• The potential has the same value at all points. 
• No energy is required to move on the same equipotential line; therefore, work is done.
• For gravity, the potential energy of each kg of mass decreases as the mass travels in the direction of the gravitational field, and increases as it travels opposite the direction of the gravitational field. 
• A positive charge will go from higher to lower potential energy along the direction of the electric field.