Let’s explore the electric potential inside the capacitor by looking at several different, but related, ways that the potential can be represented graphically.

These four graphical representations show the same information from different perspectives, and the connecting lines help you see how they are related. If you think of the elevation graph as a “mountain,” then the contour lines on the contour map are like the lines of a topographic map.

The potential graph and the contour map are the two representations most widely used in practice because they are easy to draw. Their limitation is that they are trying to convey three-dimensional information in a two-dimensional presentation. When you see graphs or contour maps, you need to imagine the three-dimensional equipotential surfaces or the three-dimensional elevation graph.

There’s nothing special about showing equipotential surfaces or contour lines every 0.5 V. We chose these intervals because they were convenient. As an alternative, FIGURE 25.20 shows how the contour map looks if the contour lines are spaced every 0.3 V. Contour lines and equipotential surfaces are imaginary lines and surfaces drawn to help us visualize how the potential changes in space. Drawing the map more than one way reinforces the idea that there is an electric potential at every point inside the capacitor, not just at the points where we happened to draw a contour line or an equipotential surface.

FIGURE 25.20 Equipotentials and electric field vectors inside a parallel-plate capacitor.

Figure 25.20 also shows the electric field vectors. Notice that

• The electric field vectors are perpendicular to the equipotential surfaces.

• The electric field points in the direction of decreasing potential. In other words, the electric field points “downhill” on a graph or map of the electric potential.

Chapter 26 will present a more in-depth exploration of the connection between the electric field and the electric potential. There you will find that these observations are always true. They are not unique to the parallel-plate capacitor.

Finally, you might wonder how we can arrange a capacitor to have a surface charge density of precisely $4.42\times {10}^{-9}\text{C}\text{/}{\text{m}}^2$. Simple! As FIGURE 25.21 shows, we use wires to attach 3.00-mm-spaced capacitor plates to a 1.5 V battery. This is another topic that we’ll explore in Chapter 26, but it’s worth noting now that a battery is a source of potential. That’s why batteries are labeled in volts, and it’s a major reason we need to thoroughly understand the concept of potential.

FIGURE 25.21 Using a battery to charge a capacitor to a precise value of $\text{Δ}{V}_{\text{C}}$.

EXAMPLE 25.7 Measuring the speed of a proton

You’ve been assigned the task of measuring the speed of protons as they emerge from a small accelerator. To do so, you decide to measure how much voltage is needed across a parallel-plate capacitor to stop the protons. The capacitor you choose has a 2.0 mm plate separation and a small hole in one plate that you shoot the protons through. By filling the space between the plates with a low-density gas, you can see (with a microscope) a slight glow from the region where the protons collide with and excite the gas molecules. The width of the glow tells you how far the protons travel before being stopped and reversing direction. Varying the voltage across the capacitor gives the following data:

Voltage (V)	Glow width (mm)
1000	1.7
1250	1.3
1500	1.1
1750	1.0
2000	0.8

What value will you report for the speed of the protons?

MODEL The system is the proton plus the capacitor charges. This is an isolated system, so mechanical energy is conserved.

VISUALIZE FIGURE 25.22 is a before-and-after pictorial representation of the proton entering the capacitor with speed $v_{\text{i}}$ and later reaching a turning point with $v_{\text{f}}=0\text{m}\text{/}\text{s}$ after traveling distance $s_{\text{f}}=\text{glow width}$. For the protons to slow, the hole through which they pass has to be in the negative plate. The s-axis has $s=0$ at this point.

FIGURE 25.22 A proton stopping in a capacitor.

FIGURE 25.22 Full Alternative Text

SOLVE The conservation of energy equation, with the proton having charge $q=e$, is $K_{\text{f}}+e{V}_{\text{f}}=K_{\text{i}}+e{V}_{\text{i}}$. The initial potential energy is zero, because the capacitor’s electric potential is zero at $s_{\text{i}}=0$, and the final kinetic energy is zero. Equation 25.28 for the potential inside the capacitor gives

$$e{V}_{\text{f}}=e\left(\frac{s_{\text{f}}}{d}\text{Δ}{V}_{\text{C}}\right)=K_{\text{i}}=\frac{1}{2}m{v}_{\text{i}}{}^2$$

Solving for the distance traveled, you find

$$s_{\text{f}}=\frac{dm{v}_{\text{i}}{}^2}{2e}\frac{1}{\text{Δ}{V}_{\text{C}}}$$

Thus a graph of the distance traveled versus the inverse of the capacitor voltage should be a straight line with zero y-intercept and slope $dm{v}_{\text{i}}{}^2\text{/}2e$. You can use the experimentally determined slope to find the proton speed.

FIGURE 25.23 is a graph of $s_{\text{f}}$ versus $1\text{/}\text{Δ}{V}_{\text{C}}$. It has the expected shape, and the slope of the best-fit line is seen to be $1.72\text{V}\text{m}$. The units are those of the rise-over-run. Using the slope, you calculate the proton speed:

$$\begin{array}{ll}{v}_{\text{i}}\hfill & =\sqrt{{\displaystyle \frac{2e}{dm}}\times \text{slope}}=\sqrt{{\displaystyle \frac{2\left(1.60\times {10}^{-19}\text{C}\right)\left(1.72\text{Vm}\right)}{\left(0.0020\text{m}\right)\left(1.67\times {10}^{-27}\text{kg}\right)}}}\hfill \\ \hfill & =4.1\times {10}^5\text{m/s}\hfill \end{array}$$

FIGURE 25.23 A graph of the data.

FIGURE 25.23 Full Alternative Text

REVIEW This would be a very high speed for a macroscopic object but is quite typical of the speeds of charged particles.

In writing the electric potential inside a parallel-plate capacitor, we made the choice that $V_{-}=0\text{V}$ at the negative plate. But that is not the only possible choice. FIGURE 25.24 shows three parallel-plate capacitors, each having the same capacitor voltage $\text{Δ}{V}_{\text{C}}=V_{+}-V_{-}=100\text{V,}$ but each with a different choice for the location of the zero point of the electric potential. Notice the terminal symbols (lines with small circles at the end) showing how the potential, from a battery or a power supply, is applied to each plate; these symbols are common in electronics.

FIGURE 25.24 These three choices for $V=0$ represent the same physical situation. These are contour maps, showing the edges of the equipotential surfaces.

The important thing to notice is that the three contour maps in Figure 25.24 represent the same physical situation. The potential difference between any two points is the same in all three maps. The electric field is the same in all three. We may prefer one of these figures over the others, but there is no measurable physical difference between them.