5.4.2: Explorations

Exploration 1: Investigate Equipotential Lines

The panel on the left displays an electric field vector plot. The arrows in the field plot represent the direction, and the colors represent the magnitude of the electric field. Restart.

1. What is an equipotential line?
2. How can such a line be determined from an electric field representation like that shown in the left panel?
3. Draw the equipotential lines for this field by dragging the pencil (at its tip) after clicking the "draw on" button. Use the print screen function on your computer to print out a copy of your drawing.
4. After you have drawn your lines, determine which potential plot \((1\:, 2,\: 3,\) or \(4)\) best represents the region. Explain your choice.
5. Was your drawing a good representation of the actual equipotential lines? Did you have any misunderstandings? Explain.

Need help? Enable Contour Lines on Double-Click to see the actual contours in the electric field view by double-clicking in the region.

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

Exploration 2: Electric Field Lines and Equipotentials

The panel on the left displays an equipotential plot. Restart.

1. What is an equipotential line?
2. How are electric field lines related to equipotential lines?
3. Draw the electric field lines for this potential by dragging the pencil (at its tip) after clicking the "draw on" button. Use the print screen function on your computer to print out a copy of your drawing.
4. After you have made your drawing, determine which electric field \((1,\: 2,\: 3,\) or \(4)\) best represents the region. Explain your choice. Note that the arrows in the field plot represent the direction, and the colors represent the magnitude of the electric field.
5. Was your drawing a good representation of the actual field? Did you have any misunderstandings? Explain.

Need help? Enable Field Lines on Double-Click to see the field lines in the contour view by double-clicking in the region.

Note

There is a short delay between the "click" and the drawing of the field line.

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

Exploration 3: Electric Potential around Conductors

In this animation you can measure electric potential using a probe. 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 dot at the current position of the probe. There are two hidden conductors in the animation. Restart.

1. How do you know when the cursor moves over a conductor?
2. Make a sketch of the animation. Begin by labeling the position of the hidden conductors on your drawing. As you find the edges of the conductors, you can use the markers to outline them (use one color of marker for the first conductor you find and the other color for the second conductor you find).
3. How could one battery be connected to the two conductors to produce the above system? What would the voltage of the battery need to be? (Hint: Remember that the zero point of potential energy is arbitrary. Does either pole of the battery have to be at \(0\text{ V}\)? Why or why not?)
4. Draw the battery in your sketch of the conductors from part (a).
5. A conductor is an equipotential surface. Why?
6. Sketch a representative number of equipotential contour lines for this system. Pick a specific voltage and then move the cursor around to find the loop of constant voltage. This maps out one equipotential contour line. As you map out other contour lines, remember that the change in voltage from one contour to the next is constant. Choose values wisely. You probably do not want to map out equipotential surfaces for voltage changes of \(0.1\text{ V}\).
7. Where is the electric field the strongest? Where is it the weakest?

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

Exploration 4: Time-of-Flight Mass Spectrometer

Positively charged particles start in the center of a uniform electric field (created by the charged gray plates; the field is shown, but fringe effects are not). When you push "play," four particles leave the parallel plates and head toward the detector. The graph simply plots the signal at the detector, showing a spike every time a particle hits the detector (position is given in centimeters and time is given in microseconds). This is a time-of-flight mass spectrometer and is used to detect what types of charged particles are in an atomic beam. Restart.

1. Given that the electric field is uniform and the voltage at the left plate is \(2000\text{ V}\) and at the right plate is \(0\text{ V}\), explain how you know that the voltage in the middle of the plates (where the particles are) is \(1000\text{ V}\).
2. How much potential energy does each charged particle have if its charge is \(1.6\times 10^{-19}\text{ C}\)? (Each need one electron to be neutral again.)
3. After each particle leaves the region with a constant electric field and enters the region without an electric field, what is the value of its potential energy?
4. What then is the value of its kinetic energy?
5. Since the particles do not have the same speeds, rank the masses of the particles from least massive to most massive.
6. By measuring the time it takes the particle to arrive at the detector and the distance the particles travel through the field-free region, determine the speed of each particle.
7. From your calculation of kinetic energy, find the mass of each particle in kilograms and atomic mass units (\(1\text{ amu} = 1.67\times 10^{-27}\text{ kg}\)).
8. Looking on a periodic table, what is the atomic mass of aluminum? It should be essentially the same as the value that you calculated for the mass of the smallest particle, as well as the mass difference between each larger particle. Therefore, this animation represents a particle beam where the first particle to hit the detector is a charged aluminum atom, and the second particle is two aluminum atoms bound together, and so forth. One way to find out what material is in an unknown substance, then, is to do this type of mass spectrometry. (Illustration 27.3 and Exploration 27.2 demonstrate other types of mass spectrometry.)

Exploration authored by Anne J. Cox.

Exploration 5: Spherical Conductor and Insulator

How does the electric potential around a charged solid insulating sphere (with charge distributed throughout the volume of the sphere) compare with the electric potential around a charged conducting sphere? Move the test charge to map out the electric potential as a function of distance from the center (position is given in centimeters and electric potential is given in volts). Restart.

1. Why is the voltage constant inside the conductor?
2. Why is there no electric field and no force on the test charge inside the conductor?
3. Looking at the plots you make of voltage as a function of radial distance (as you move the test charge), what is the same and what is different between the two cases? Given that both spheres have the same total charge, explain the similarities and differences in the plots.
4. The electric field outside both spheres is \(Q/4\pi\varepsilon_{0}r^{2}\). Using this and the reference point of \(V = 0\) volts at infinity, find an expression for the electric potential at a point outside the sphere and a distance \(r\) from the center of the sphere. \(V = -\int\mathbf{E}\cdot d\mathbf{r}\) and integrate from \(r =\) infinity (where \(V = 0\) volts) to a point \(r\).
5. Measure the voltage at some point outside the sphere and find the charge on both spheres. Verify that the total charge is the same.

Now for the voltage inside the uniformly charged insulator. Here the electric field is \(Qr/(4\pi\varepsilon_{0}R^{3}\)), where \(R\) is the radius of the sphere itself. In this case, to find the potential as a function of \(r\), you again need to integrate \(V = -\int\mathbf{E}\cdot d\mathbf{r}\), but this time you must break up the integral and integrate from infinity to \(R\) using \(E = Q/4\pi\varepsilon_{0}r^{2}\) (to find the electric potential associated with getting all the charges to the surface of the sphere) and then integrate from \(R\) to \(r\) (an arbitrary point inside the sphere) using the expression for the electric field inside the insulating sphere.

6. Verify that your calculation gives the same results as shown on the graph.

Exploration authored by Anne J. Cox.
Script authored by Mario Belloni 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.