5.5.3: Problems

Exercise \(\PageIndex{1}\): Capacitor dimensions (identify the correct graph)

As you move the slider, you change either the area or the separation of a capacitor (position is given in centimeters and capacitance is given in picofarads). Restart. Which of the graph(s) is (are) correct? Explain.

Problem authored by Morten Brydensholt.

Exercise \(\PageIndex{2}\): Describe the charge distribution and find the electric potential across a capacitor

A parallel-plate capacitor is shown. A battery connects the two plates together but is not shown. Position and electric field strength are shown when you drag the mouse (position is given in meters and electric field strength is given in newtons/coulomb). Restart.

1. Is the green plate positively charged, negatively charged, or neutral? Explain.
2. Is the purple plate positively charged, negatively charged, or neutral? Explain.
3. Sketch a diagram of the charge distribution on the plates.
4. What is the voltage of the battery that connects the two plates?

Problem authored by Melissa Dancy and modified by Anne J. Cox.

Exercise \(\PageIndex{3}\): Is this capacitor connected to a battery?

Move the slider to change the area of the two capacitor plates (position is given in centimeters, area is given in centimeters squared, and energy is given in microjoules). Restart. Is this a capacitor connected to a battery or a charged capacitor that is not connected to a battery? Explain.

Problem authored by Anne J. Cox.
Script authored by Morten Brydensholt and modified by Anne J. Cox.

Exercise \(\PageIndex{4}\): Find the dielectric

The animation shows a parallel-plate capacitor with a hidden dielectric between the two plates. You can click-drag to measure position and electric potential (position is given in meters and electric potential is given in volts). The small black circle in the center is a dragable test charge that can be used to show the direction and magnitude of the electric field. Click on the add marker links to add a blue or red marker at the location of the test charge to outline the edges of the dielectric. Restart.

1. Sketch and label an estimate of the location of the dielectric.
2. Sketch a representative number of equipotential lines.

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

Exercise \(\PageIndex{5}\): Rank the value of the dielectrics

A parallel-plate capacitor has three dielectrics inserted between the plates. The battery connecting the two plates is off screen. The blue and red circles on the plates represent the accumulated charge. Position and electric field strength are shown when dragging the mouse (position is given in meters and electric field is given in newtons/coulomb). The arrows represent the electric field vectors. Restart.

Rank the dielectrics based on their dielectric constants, from smallest to greatest. Explain your ranking.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{6}\): Rank the value of the dielectrics

A parallel-plate capacitor has three dielectrics inserted between the plates (position is given in meters and electric potential is given in volts). A battery that connects the two plates is off screen. The blue and red circles on the plates represent the accumulated charge. The lines represent equipotential surfaces. Restart.

Rank the dielectrics based on their dielectric constants, from smallest to greatest. Explain your ranking.

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

Exercise \(\PageIndex{7}\): Which capacitor with dielectric is connected to a battery?

The animations show a parallel-plate capacitor with a movable dielectric while the plots show the electric potential (left plot) and charge (right plot) as a function of the \((x,\: y)\) position (position is given in meters and the electric field strength is given in newtons/coulomb). You can click-drag your mouse in a graph to rotate a plot and see it from a different angle. You can move the dielectric into and out of the region between the two plates by clicking and dragging the middle of the dielectric. Restart.

1. In one of the animations a capacitor is connected to a battery, and in the other one a charged capacitor is not connected to the battery. Which is which?
2. Describe the changes you see on the plots in the two animations and explain what is happening as the dielectric is dragged between the plates (i.e., why do you see the increases or decreases in potential or charge?).

Note

If the animation says "Did not converge" after you move the dielectric (especially if you move it from one end to another), simply click on the dielectric again to have the animation recalculate.

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

Exercise \(\PageIndex{8}\): Find the value of the dielectric constant

The animation shows a parallel-plate capacitor that has been charged and then disconnected from a battery so that the charge on the plate remains constant. The red and blue circles inside the plates represent the charge build up on the plate and the electric field vectors are also shown (position is given in meters, electric field strength is given in newtons/coulomb and electric potential is given in volts). A dragable dielectric (drag at the center) is below the capacitor. What is the value of the dielectric constant of the dielectric? Restart.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{9}\): Capacitors in parallel (microscopic view)

The animation shows two parallel-plate capacitors connected by conducting wires to a battery (the two circles to the left). When you push the "play" button, the battery is connected to the capacitors and electrons are pulled away from the associated positive charges (charge separation occurs). Restart.

1. Which capacitor has the larger capacitance?
2. Which has the larger electric potential difference between the plates?
3. Which has the greater electric field between the plates? Explain.

Problem authored by Anne J. Cox.
Script authored by Morten Brydensholt.
Applet authored by Vojko Valencic.

Exercise \(\PageIndex{10}\): Capacitors in series (microscopic view)

The animations show parallel-plate capacitors connected by conducting wires to a battery. Restart. If all the capacitors are identical, how do the batteries in the two cases compare?

Problem authored by Anne J. Cox.
Script authored by Morten Brydensholt.
Applet authored by Vojko Valencic.

Exercise \(\PageIndex{11}\): Equivalent capacitance: what's wrong with this circuit?

In the animation, you can close and open switches and read the voltage across capacitors \(A,\: B,\) and \(C\). The capacitance of capacitor \(C\) is \(1\times 10^{-5}\text{ F}\), and the voltage of the battery is \(10\text{ V}\). Restart. What, if anything, is wrong with the animation?

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

Exercise \(\PageIndex{12}\): Calculate capacitance of concentric spheres

The animation shows a spherical conductor surrounded by a conducting spherical shell. The two conductors have equal but opposite charge and the voltage difference between the two is \(50\text{ V}\) (position is given in centimeters and charge is given in nanocoulombs). The field vectors for the electric field (\(kQ/r^{2}\)) are shown in the region between the shell and the inner sphere and are zero everywhere else. Restart.

1. Develop an expression for the charge as a function of the radius of the outer sphere and verify it with the animation.
2. Develop an expression for the capacitance as a function of radius.
3. Does the capacitance increase or decrease with increasing shell size? Explain.

Problem authored by Anne J. Cox.

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