# 2.1: Background Material

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- Tom Weideman
- Lecturer (Physics) at University of California, Davis

## Text References

## Potentials in General

It's quite possible that you are taking on this week's lab without having seen the topic of electric potential in lecture. Given that this is a lab about electric potential, this might seem to pose a problem, but a brief review of the concept of potential energy from Physics 9A (see the first text reference above) should be sufficient to get you where you need to be for this lab.

In Physics 9A, we linked the ideas of potential energy and force. We found that the force vector at a point in space is determined from the *gradient *of the potential energy function at that point. In conceptual terms, it works like this: Place yourself at a point in space, and note the potential energy at that position. Then turn around and around, noting the nearby potential energies that you see in every direction. Whichever direction points to the nearby potential energy that is the biggest drop from where you are positioned, that is the direction that the force is exerted.

If you turn to look in a direction in which the potential energy doesn't change at all, then there is no component of the force in that direction. If you start traveling around in the space, always moving in directions where the potential remains the same, then your motion is confined to something called an *equipotential surface*. With no component of the force acting parallel to the equipotential surface, the total force vector must be perpendicular to it.

Now in Physics 9C, we are using language we did not use in Physics 9A. What we called a "potential energy function" we now refer to as a "scalar field." Note that it means the very same thing – every point in space has associated with it a scalar number. The force vectors we derived from gradients of this field are different at every point in space, so with a vector defined at every point in space, we refer to that as a "vector field." The only other leap we take in Physics 9C is that we use a scalar field that is not potential energy, and a vector field that is not force. Instead, they are electrostatic potential and electric field, respectively. They have different physical meanings and are measured with different units than their 9A counterparts, but the basic ideas are exactly the same: There are equipotential surfaces – surfaces of equal electrostatic potential (voltage) – and a vector (electric) field that is perpendicular to those surfaces everywhere.

## Mapping Equipotential Surfaces

This week's lab involves a voltage supply connected to a pair of conductors that can come in different shapes. One conductor will be held at a lower potential, and the other at a higher potential. This potential difference will produce an electric field between the two conductors, which is manifested as a continuously-varying electric potential in the space separating them.

We will use an apparatus called a *multimeter* to measure the potential difference between to points. It can't do this in air, but rather the probes that come out of this device have to be in contact with solid objects from (and into) which charge can flow.

**Figure 2.1.1 – Parallel Conducting Plates**

In the figure above, we have two parallel conducting plates. The red and black wires are coming from the positive and negative leads respectively of a voltage supply (essentially a battery) of 5 volts. The green and yellow wires connected to the same conductors come from the multimeter, which has been set to measure voltage differences. Note that the digital display confirms that the voltage difference between the two plates is 5 volts.

Our goal is to map out the potential at points between these two plates, but how do we do this if the multimeter can only register voltage differences between solid objects, how do we measure the potential in the space between the plates? Well, it turns out that the black paper atop which the conductors are placed actually conducts electricity, so we can touch a multimeter probe to different points on the paper to measure a potential difference. Leaving the yellow wire where it is, we arbitrarily define that conductor's potential to be zero. We can touch the green probe to this conductor and the multimeter confirms that it is at a voltage difference of zero with the yellow probe:

**Figure 2.1.2 – Top Plate Defined as Zero Potential**

Now using the green probe to test various points on the black paper between the two plates will give us a map of the equipotential surfaces that exist between them. In particular, we will search of the equipotential surfaces with potentials of 1, 2, 3, and 4 volts between the plates:

**Figure 2.1.3 – Finding Equipotentials Between the Plates**

To record the positions of these potentials, we sandwich a piece of carbon paper between the black conducting paper on top, and a white sheet of paper below. Then bearing down on the green probe creates a dot on the white paper. After doing this at several positions for each potential, a picture of the equipotential surfaces emerges.

**Figure 2.1.4 – Equipotentials Between Parallel Plates**

The solid lines on the top and bottom were traced from the conductors, which means they are slightly closer together than the conductors are, so it is probably best to ignore them. We note that the dotted lines formed by the data points are approximately equally-spaced, which is what we would expect for parallel-plates, because between parallel plates we expect to see a uniform electric field. Calling the bottom line the \(x\)-axis (i.e. \(y=0\)), we have:

\[E=-\overrightarrow\nabla V \;\;\;\Rightarrow\;\;\; E_y= -\dfrac{\partial}{\partial y}V\;\;\;\Rightarrow\;\;\;V\left(y\right)=-E_yy+const\]

With \(E_y\) a constant value, the potential varies linearly with position, which explains why the equipotentials are equally-spaced. Plugging-in a value of 5 volts for the potential at \(y=0\) gives the integration constant, and measuring distances allows us to compute the magnitude of the uniform electric field.

Finally, it should also be noted that the equipotentials nearest to the conductors bend somewhat near the edges. This demonstrates the "fringe effects" that occur near the edges of the conductors, where the approximation of a uniform field breaks down.