Electric Potential Energy and Work

In the three previous pictures is a setup of two light bulbs connected in parallel to a power source. It is clear to see that the bulbs glow with different intensities. This indicates the power supplies to these two bulbs are different, proving that potential is not constant in across a circuit in parallel.

In an experiment where we analyzed the effects that a metal conductor connected to a source would have on the temperature of a cup of water, we see that the temperature increases steadily over time.

When the voltage is increased, the temperature rises faster.

Depicted on the left side are two lights bulbs in parallel with a power source. The lower image is the accepted representation of this setup that would be used by engineers. On the right side is the two bulbs in series connected to two batteries in parallel. The right side has a voltage equivalent to one battery. 

Doubling the voltage essentially doubles the current running through a conductor. Therefore, increasing the voltage by a factor of two, increases the current by a factor of two, which increases the power by a factor of four.

Calculating the work done to get from point a to point c, we determined that only movement parallel to the gravitational field performs work (in regards to gravity) on the object resulting in a change in gravitational potential energy. The same can be said about the change in potential energy on a charge in an electric field. Only movement parallel to the field results in change in the field. 

Depicted here is an equipotential surface. At all points on the circle, there is a constant potential. If we integrate from infinity to a distance r away from the charge, we see that the potential at r is equal to (k*Q)/r.

In the previous two pictures, we show how to calculate the potential at a point due to numerous charges. As you can see, the total potential is simply the sum the potentials due to each charge.