5.2.1: Illustrations

Illustration 1: What is an Electric Field?

This animation plots a vector field when you enter values for the \(x\) component and \(y\) component of the field. You should try several values to get a sense of what a vector field is. Restart.

Begin by creating a simple uniform vector field by entering \(5\text{ N/C}\) for \(E_{x}\) and updating the field. Notice that the animation displays a grid of arrows pointing to the right. If you enter \(-5\text{ N/C}\) for \(E_{x}\), the field arrows will point in the opposite direction. Enter \(3\text{ N/C}\) for \(E_{x}\) and \(4\text{ N/C}\) for \(E_{y}\) and update the field again. The arrows now point at an angle of \(37\) degrees with respect to the \(x\) axis. Now, if you enter \(2\text{ N/C}\) for \(E_{x}\), what do you see? How is it different from \(5\text{ N/C}\) for \(E_{x}\)? What does the color of the vector show? Why do you think we do not represent magnitude of the vector field by the length of the vector?

Now, build a field that you are familiar with (whether you know it or not) by putting in \(0\text{ N/C}\) for \(E_{x}\) and \(-4.9\text{ N/C}\) for \(E_{y}\). This is a representation of the vector force field for a \(0.5\text{-kg}\) mass close to Earth's surface. Why? What would the vector force field be for a \(3\text{-kg}\) mass close to Earth's surface? (If it is far away from Earth's surface, like a satellite in orbit, you need to take into account the decrease in gravitational attraction as a function of distance squared).

The values of the field components do not need to be constants. Try \(2\ast x\) for \(E_{x}\) and \(2\ast y\) for \(E_{y}\). What do you see? In this case, the vectors show you a field that changes in both magnitude and direction with position. For \(x = 0\text{ m}\), \(y = 2\text{ m}\), what are the values of \(E_{x}\) and \(E_{y}\)? Does the arrow on the screen point in the correct direction at that point? Repeat this exercise with \(2\ast y\) for \(E_{x}\) and \(2\ast x\) for \(E_{y}\).

Try some other set of (nonconstant) values for \(E_{x}\) and \(E_{y}\). Specifically, try \(E_{x} = x/(x\ast x + y\ast y)\wedge 3/2\) and \(E_{y} = y/(x\ast x + y\ast y)\wedge 3/2\). What does this vector field look like?

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

Illustration 2: Electric Fields from Point Charges

This Illustration allows you to add charges by clicking on the appropriate link. All charges are added at the center of the animation; you must drag charges from the origin to see the effect of subsequent charges. Restart.

First, examine the field around one charge. What does the field look like for a \(1\text{-C}\) charge? Clear the charge and add a \(2\text{-C}\) charge. How is that field configuration different? Clear the charges and then add a \(-2\text{-C}\) charge. What is the difference? Notice that the strength of the field is represented by the color of the field vectors. White is the smallest magnitude (zero) and black is the greatest, with blue, green, and red in between. When the charge is negative, the electric field vectors change direction and now point in the opposite direction. Positive charges have field vectors that point radially outward, and negative charges have field vectors that point radially inward.

Clear the charges and add two positive charges of the same magnitude. Notice that since the charges are added at the center, you must drag a charge away to see the one underneath. How is the field different with two charges compared with one? Move one of the charges closer and farther away from the other one. When the charges are sitting on top of each other, what does the field look like? When you move them far apart, what does it look like? Notice that the fields add together (it is nothing more than vector addition). The fact that the electric field at any point is the vector sum of the electric fields due to the surrounding charges is simply the principle of superposition. You have seen this in the previous chapter: The force on a charge is due to the sum of the Coulomb forces from the surrounding charges. Notice that the force vector of an individual charge points in the direction of the electric field due to the other charge. It does not, however, point in the direction of the electric field due to both charges. The field configuration shown would be the field experienced by a third particle (not the force experienced by either particle).

What do you predict the field will look like with two negative charges (of equal magnitude)? Try it. What are the similarities and differences between the two positive and two negative charge distributions? The field vectors point in the opposite direction.

What about a dipole, one positive and one negative charge? How is it the same or different from two charges of the same sign? What is the direction of the field at the midpoint between the charges? The vector field can be described in terms of the vector sum of the field from the two particles.

Try two charges of different magnitude. What does the field look like? Notice that there is a point at which the electric field is zero directly in between the two charges. If you added a third charge at that spot, what do you predict the force on it would be? Try it. Notice that the force on the third charge is simply due to the electric field from the other two charges (multiplied by the charge of the third charge).

Add three or four charges and look at the field. Pick one point of the electric field and explain why it points in the direction it does. How can you tell, simply by looking at the field (and not the labels on the charges), which ones are positive and which ones are negative? How can you tell which ones have more charge?

Illustration authored by Anne J. Cox and Melissa Dancy.

Illustration 3: Field-Line Representation of Vector Fields

There are different ways to represent the electric field created by a charge distribution. One way is to use field vectors (as you've already seen), but you may find it a bit tedious (and difficult unless you carry around a colored pencil set) to draw that on your paper. Many books use electric field lines as an alternate representation of field vectors.

Switch between the field-vector and the field-line representation for Configuration A. What is the difference between the two representations? In a field-line drawing, the line density is often used to represent, at least qualitatively, field strength (more field lines in an area indicate a larger electric field). The arrows represent the direction of the electric field. Now move the charges around in Configuration A. How does the field-line representation reflect the change? Pick a point on a field line. Switch to the vector field representation. What does the field vector look like at that point? Notice that the field vector points in a direction tangent to the field line at any point.

Now, consider Configuration B and look at the two representations. Can you tell if the net charge distribution is positive, negative, or zero? Move the charges to check your answer (you can put them all on top of each other).

Illustration authored by Anne J. Cox and Melissa Dancy.

Illustration 4: Practical Uses of Charges and Electric Fields

When you first play this animation, there is no electric field present (position is given in centimeters and time is given in seconds). Input values for the electric field in the \(y\) direction and run the animation again. You will notice that the charge is deflected from its original straight-line path. Restart. Consider what happens when you

• increase the charge.
• increase or decrease the initial velocity.
• change the sign of the electric field.

Can you make the charge hit the green target? What happens if an object of zero charge is shot into a region where there is an electric field? The simple ideas demonstrated in the animation have been used in several real world applications.

Ever wonder how your computer and television monitor (those that are not flat-panel LCD or plasma) work? The image you see on the screen is produced using a cathode ray tube (CRT). The CRT uses a heated filament (not unlike that found in a light bulb) to produce electrons traveling at high velocities. The electron is shot into a region of constant electric field. Since the electric field exerts a force on charges, the electron is deflected from its path. The amount of deflection depends on the strength of the electric field and the velocity of the electron before it encountered the field. By controlling either the initial velocity or the field strength, the spot where the charge hits the screen is controlled. In the case of the CRT, the charge is not varied since it is a stream of electrons that is produced. The screen is coated with a substance (phosphor) that will emit light when struck by an electron. The rate at which electrons are striking the screen is very fast, allowing a complete picture to be built up. The applet shows electrons being deflected up and down. Notice that another set of plates could be added to control the right and left movement of the electron. Most new CRTs use magnetic fields to control the electron, but the basic idea is the same. Since the location where the electron hits the screen is directly related to the strength of the electric field, a CRT can be used to measure the electric field. Oscilloscopes use this idea to measure voltages (by measuring the electric fields produced), and display time-dependent voltage information on a screen.

Years ago, most printing was accomplished by impacting something mechanical against the paper. Typewriters and dot-matrix printers both used this concept. Today, high-quality, versatile printing is often done using an ink-jet printer. If you have a printer connected to your home computer, it is probably an ink-jet. The ink-jet printer works by squirting tiny drops of ink on the paper. These drops are very small, with a diameter less than a human hair. The number of dots a printer can place in an inch is specified by the dpi (dots per inch) number and is often around \(1200\) or more in the horizontal direction. There are two technologies used to drop the ink on the paper: continuous-ink printing and forced-ink printing. Forced-ink printing is the most common today and utilizes some method of forcing the ink to drop when it is directly over the desired location. From a physics point of view, the continuous-ink printing method is more interesting because it controls the location of the ink drop with an electric field.

When an ink drop is ejected from the ink cartridge, it is given a computer-controlled charge by the printer and then passed through an electric field. The location where the drop lands on the paper is determined by the charge on the drop. The same basic idea that is used to light up a particular portion of your television or computer screen can also be used to place ink on a particular location of your paper.

Illustration authored by Melissa Dancy.

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