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# 2.1: Background Material

## 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.