5.5.2: Explorations

Exploration 1: Energy

Wait for the calculation to finish. As you move the bottom capacitor (and allow the applet to finish calculating after each move), the graph shows the stored energy as a function of separation distance between the plates (position is given in millimeters, charge is given in nanocoulombs, and energy is given in nanojoules). Restart.

1. Given that the stored energy (the potential energy) is \(QV/2\), what is the voltage difference between the plates?
2. Does the voltage difference between the plates change?
3. How does the capacitance change as you move the plates?
4. What is the area of the plates for this capacitor?
5. Why does the charge change as you move the plates?
6. As you move the plates closer together, does the stored energy increase or decrease?
7. Does that mean you would need to do positive work to push the plates together or pull them apart? Explain.
8. Since potential energy \(U = QV/2\), if \(V\) is kept constant, what is \(U\) (potential energy) as a function of the separation distance? Verify that this is the relationship shown on the plot.

Exploration authored by Anne J. Cox.

Exploration 2: Capacitors, Charge, and Electric Potential

This animation shows a parallel-plate capacitor and the charges on the plates, the total charge, and the electric potential difference between the plates. You can move the left plate by click-dragging the middle of the plate (at the "Drag Me" label). The plots show you the electric potential and charge as a function of \((x,\: y)\) position (position is given in centimeters, charge is given in coulombs, electric field strength is given in newtons/coulomb, and electric potential is given in volts). You can click-drag in a graph to rotate the plot and see it from a different angle. Restart.

1. Which plot corresponds to electric potential as a function of position? Which one is charge as a function of position? Explain how you know.
2. From the charge plot, where is there the most charge on the plates? Why?

Consider the configuration with a constant electric potential difference between the plates.

3. How does the charge change as you move the left capacitor plate? Explain.
4. Does this result correspond to a capacitor connected to a battery or a charged capacitor disconnected from a battery? Explain.

Now consider the configuration with a constant charge on the plates.

5. How does the voltage difference between the plates change as you move the left capacitor plate? Explain. (Note: If the animation says "Failed to converge" after you move the plates, simply click on the plate again to have the animation recalculate. The charge will stay in about the same range.)
6. Does this correspond to a capacitor connected to a battery or a charged capacitor disconnected from a battery? Explain.

Exploration authored by Anne J. Cox.
Script authored by Morten Brydensholt.

Exploration 3: Conductors and Dielectrics

Wait for the calculation to finish. There are hidden conductors and dielectrics in the animation. The light red circles represent positive charges, and the light blue circles represent negative charges. This charge can be either bound or unbound (free); in other words it could be the charge on a dielectric or on a conductor. You can measure electric potential using the probe and you can click-drag to measure position and electric potential (position is given in meters and electric potential is given in volts). Clicking an "add marker" link will add a circular marker at the current position of the probe. Restart.

1. Sketch and label your best guess as to the configuration of the hidden conductors and dielectrics. You may want to use the markers to outline conductors and/or dielectrics.
2. What is the minimum number of external batteries needed to produce the system? Show the voltage values of these batteries and how they should be connected to the system.
3. Where is the electric field strongest? Where is it weakest?
4. Sketch the electric field using electric field lines; that is, draw a representative number of field lines.
5. Sketch the equipotential lines.

Problem authored by Mario Belloni, Wolfgang Christian and Melissa Dancy.

Exploration 4: Equivalent Capacitance

This animation contrasts two configurations of capacitors and a battery (capacitance is given in farads). The table shows the voltage across the battery as well as across each capacitor. Restart.

First, consider capacitors in series. Pick a value of the capacitance for capacitor \(A\) that is bigger than capacitor \(B\).

1. What is the charge on each capacitor (use \(Q = CV\))?
2. Why are the charges equal (explain in terms of where the charge originates to charge the plates)?

This is the total charge stored in this circuit. If the battery were removed from this circuit and we wanted to use the stored charge for an electrical appliance, notice that the charges stored on the two sides of the capacitors connected to each other (the bottom plate of \(A\) and the top plate of \(B\)) would not be available to another circuit. Thus, the total charge stored is the charge stored on either capacitor.

3. If you wanted to replace the two capacitors in this circuit with one capacitor that stored the same amount of charge at the same total voltage, what would the value of that capacitance be?
4. Verify that the equivalent capacitance is equal to \((1/C_{A} + 1/C_{B})^{-1}\).

Now consider capacitors in parallel. Pick a value of the capacitance for capacitor \(A\) that is different from capacitor \(B\).

5. What is the same for the two capacitors? What is different?

This time, the charge stored on each capacitor would be available to an electrical appliance if the battery were removed; thus, the total charge is the sum of the charge stored individually on each plate.

6. What is the equivalent capacitance for these two capacitors (i.e., what size capacitor would store the same total charge at this voltage?)?
7. Show that it is equal to \((C_{A} + C_{B})\).

Exploration authored by Anne J. Cox.
Applet authored by Toon Van Hoecke.

Exploration 5: Capacitance of Concentric Cylinders

Wait for the calculation to finish. This animation shows a coaxial capacitor with cylindrical geometry: a very long cylinder (extending into and out of the page) in the center surrounded by a very long cylindrical shell (position is given in centimeters, electric field strength is given in newtons/coulomb, and electric potential is given in volts). The outside shell is grounded, while the inside shell is at \(10\text{ V}\). You can click-drag to measure the voltage at any position. Restart.

1. Use Gauss's law to show that the magnitude of the radial electric field between the two conductors for a cylindrical coaxial capacitor of length \(L\) is \(E = Q/2\pi rL\varepsilon_{0} = 2kQ/(rL)\), where \(Q\) is the total charge on the inside (or outside) conductor and \(r\) is the distance from the center.
2. If \(L = 1\text{ m}\), measure the electric field in the region between the two conductors and determine the charge on the inside (and outside) conductor.
3. Use \(V = -\int\mathbf{E}\cdot d\mathbf{r}\) to show that the potential at any point between the two conductors is \(V = (Q/2\pi L\varepsilon_{0}) \ln(b/r)  = (2kQ/L) \ln(b/r)\), where \(b\) is the radius of the outer conductor.
4. Given that the potential difference between the two cylinders is \(10\text{ V}\), verify your answer to (b) and find the charge on each conductor.
5. Given, then, that the potential difference between the two conductors is \(V = (Q/2\pi L\varepsilon_{0}) \ln(b/a) = (2Qk/L) \ln(b/a)–b\) is the radius of the outer shell and a is the radius of the inner cylinder—show that the capacitance of this capacitor is \((2\pi L\varepsilon_{0})/ \ln(b/a) = (L/2k)/ \ln(b/a)\).
6. What is the capacitance (numerical value) of this capacitor?

Exploration authored by Anne J. Cox.
Script authored by Mario Belloni, Wolfgang Christian and Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.