Check out the equipotential surfaces of a bunch of charges dancing around each other.
Lecture 2 | EP Field Patterns & Motion of Charges
Electric Potential
Lecture 2 | EP Field Patterns & Motion of Charges
Electric Potential
Pre-lecture Study Resources
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
Electric Potential | Charges in a Field and the Total Energy of a System of Charges
Positive charges feel a force and speed up in the direction of decreased electric potential.
It's because that decreases the electric potential energy. All particles feel a force in the direction of decreased potential energy, but be careful, for a negative charge, it feels a force toward increased electric potential because that actually decreases the electric potential energy. The minus sign on the charge accounts for the difference between the positive charge moving towards decreased potential while the negative charge moves towards increased potential. Both do so to decrease the potential energy.
This example also illustrates the power of the field concept. Once we have found the electric potential field from a charge distribution, we can perform an infinite number of theoretical experiments on how much energy is converted between kinetic and potential for of any charge moving between locations in space. To make it more tangible, lets connect to the analogy of the gravitational field and gravitational potential energy. Recall that gravitational potential energy $U^g = mgh$. The gravitational field is equal to $gh$ and when multiplied by the mass you get the gravitational potential energy $mgh$. Since $g$ is a constant, the field really only depends on your height (or elevation). A topographical map is essentially a gravitational potential map.
This is an image of comparing a voltage map to a topographic map. On the voltage map there are diagonal equipotential lines which are similar to a topographic map where the lines show equal levels or elevations.
Lines (or really surfaces) called equipotentials can be used to describe places that have a constant potential and thus energy. Masses feel a force that make them want to move down the equipotentials, analogous to the positive charge above feeling a force towards decreased electric potential. In both cases (charges and masses) the systems move towards decreased potential energy. This is one of the most fundamental statements that can be made in physics - systems drive towards decreased potential energy. This is why water runs downhill.
Total Potential Energy of a Collection of Particles
One question that often arises is what is the total electric potential energy of a collection of particles. Imagine a single positive charge fixed in space. To bring a second charge close to the first, you'd have to do work to overcome the repelling force. If you did overcome that work and brought them next to each other, if they were released, they would fly away with kinetic energy. That kinetic energy comes originally from the work of bringing them together, which is converted into potential energy when they are close together. That potential energy can be thought of as the energy of the system. If you where to bring in more charges, you'd have to figure out the potential energy of all the combinations particles. The total potential energy for a system of three particles is described below.
This is an image titled total potential energy of a system: energy to bring all q’s together. It shows three charged ions in a triangular formation. The first is positively charged labeled q one on the bottom left and the second is negatively charged labeled q two on the top right and directly below is a positively charged ion called q three. It shows three different equations of the different potential forces acting on each particle. The first equation is the potential energy of one and two which is equal to the voltage of one at r two multiplied by q two which is also equal to k multiplied by q one and q two all divided by the distance between ion one and two. The appropriate equations are also written for two and three as well as one and three.
The total potential energy of the system of three particles is just summation of all the combinations. *Warning* don't over count interactions, $U_{12}$ is the total energy of the 1 and 2 system, meaning if you were to also include $U_{21}$, you'd be double counting. So it's only the unique pairs and the ordering of the subscripts does not matter.
$U_{total}=U_{12} + U_{23} +U_{13}$ ... + all combinations if there are more particles
This is how much energy would be released into kinetic energy if the system was released from rest. Notice that if there were negative charges present you could have a negative total potential energy. Systems with a negative potential energy are said to be bound systems as the particles have a net attraction to one another. They want to clump together as opposed to fly apart.
Key Equations and Infographics
Now, take a look at the pre-lecture reading and videos below.
Required Videos
OpenStax Section 19.4 | Equipotential Lines
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