Topic 6.2 Extended B – Electric field

1. FYI: School A must assume that the force "signal" travels infinitely fast, to react to relative motion between the masses. Topic 6.2 ExtendedB – Electric field FYI: School B assumes that the force "signal" is already in place in the space surrounding the source of the force. Thus the force signal does NOT have to keep traveling to the receiving mass. The normal force, the friction force, tension and drag are all classified as contact forces, occurring where two objects contact each other. FYI: Relativity has determined that the absolute fastest ANY signal can propogate is c = 3.0108 m/s, the speed of light. The gravitational force, and the electric force, however, do not need objects to be in contact (or even close proximity). These two forces are sometimes called action at a distance forces. There are two schools of thought on action at a distance. School A: The masses know where each other are at all times, and the force is instantaneously felt by both masses at all times. School B: The masses deform space itself, and the force is simply a reaction to the local space, rather than the distant mass.

2. SUN Topic 6.2 ExtendedB – Electric field Thus School B is currently the "correct" view. If, for example, the gravitational forces were truly "action at a distance," the following would occur: c c The planet and the sun could not "communicate" quickly enough to keep the planet in a stable orbit.

3. FYI: You have probably seen such a model at the museum. A coin is allowed to roll, and it appears to be in orbit. FYI: Of course, if the slope isn't just right to match the tangential speed, the coin will spiral into the central maw. Topic 6.2 ExtendedB – Electric field Instead, look at the School B view of the space surrounding the sun: The planet "knows" which way to "roll" because of the local curvature of the space surrounding the sun. FYI: The view of School B is called the FIELD VIEW. Long distance communication is not needed.

4. m0 m0 Topic 6.2 ExtendedB – Electric field Of course, if the mass isn't moving, it will roll downhill if placed on the grid: Depending on where we place the "test" mass m0, it will roll differently. Question: Why is the arrow at the location of the first test mass smaller than that at the second one?

5. Topic 6.2 ExtendedB – Electric field We can assign an arrow to each position surrounding the sun, representing the direction the test mass will go, and how big a force the test mass feels. We represent fields with vectors of scale length. In the case of the gravitational field, the field vectors all point toward the center:

6. SUN Topic 6.2 ExtendedB – Electric field What the field arrows tell us is the magnitude and the direction of the force on a particle placed anywhere in space: If we view our gravitational field arrows from above, we get a picture that looks like this: The blue particle will feel a "downward" force. The red particle will feel a "leftward" force whose magnitude is LESS than that of the blue particle. FYI: We don't even have to draw the object that is creating the field.

7. - SUN FYI: Inward-pointing fields are called SINKS. Think of the field arrows as water flowing into a hole. Topic 6.2 ExtendedB – Electric field FYI: Test charges are by convention POSITIVE. Therefore, field vectors around a charge show the direction a POSITIVE charge would want to go if placed in the field. Consider, now, a negative charge: For masses, all we have is an attractive force. If we place a very small POSITIVE test charge in the vicinity of our negative charge, it will be attracted to the center: All of the field arrows point inward. We can map out the field vectors just as we did for the sun.

8. + - FYI: Outward-pointing fields are called SOURCES. Think of the field arrows as water flowing out of a fountain. FYI: We call such a system an ELECTRIC DIPOLE. A dipole has two poles (charges) which are opposite in sign. Topic 6.2 ExtendedB – Electric field Question: T or F: Gravitational fields are always sinks. FYI: The ELECTRIC DIPOLE consists of a source and a sink. Now suppose our charge is positive. We can place a negative charge here. Question: T or F: Electric fields are always sinks. + - The fields of the two charges will interact: If we place a very small POSITIVE test charge in the vicinity of our positive charge, it will be repelled from the center: Keep in mind that the field lines show the direction a POSITIVE charge would like to travel if placed in the field. We can map out the field vectors just as we did for the negative charge. A negative charge will do the opposite:

9. + - strong weak Topic 6.2 ExtendedB – Electric field You may have noticed that in the last field diagram some of our arrows were unbroken. We can draw solid field arrows as long as we note that THE CLOSER THE FIELD LINES ARE TO ONE ANOTHER, THE STRONGER THE FIELD. FYI: There are some simple rules for drawing these solid "electric lines of force." Rule 1: The closer together the electric lines of force are, the stronger the electric field. Rule 2: Electric lines of force originate on positive charges and end on negative charges. Rule 3: The number of lines of force entering or leaving a charge is proportional to the magnitude of that charge.

10. C D A B E F Question: Which charge is the strongest negative one? D Question: Which charge is the weakest positive one? E Topic 6.2 ExtendedB – Electric field Question: Which charge is the strongest positive one? A Determine the sign and the magnitude of the charges by looking at the electric lines of force. Question: Which charge is the weakest negative one? C

11. Electric field definition Electric field in space surrounding a point charge Topic 6.2 ExtendedB – Electric field So how do we define the magnitude of the electric field vector E? Here's how: F on qo E = q0 where q0 is the positive test charge Think of the electric field as the force per unit charge. To obtain a more useful form for E, consider the force F between a test charge q0 and an arbitrary charge q: kq0q r2 F = kq0q q0r2 kq r2 F on qo = E = then = q0 so that kq r2 E =

12. Force on a charge placed in an electric field Topic 6.2 ExtendedB – Electric field What is the magnitude of the electric field strength two meters from a +100 C charge? Use kq r2 E = 9109·10010-6 22 = = 225000 n/C What is the force acting on a +5 C charge placed at this position? From F on qo we see that F = qE E = q0 F = qE so that = 510-6·225000 = 1.125 n

13. Topic 6.2 ExtendedB – Electric field EB A What is the electric field at the chargeless corner of the 2-meter by 2-meter square? +10 C EA EA EC -10 C +10 C C B Start by labeling the charges (organize your effort): kq r2 kq r2 kq r2 |EC| = |EA| = |EB| = 9109·1010-6 22 9109·1010-6 22 9109·1010-6 ( 22+22 )2 = = = = 22500 n/C = 22500 n/C = 11250 n/C Now sketch in the field vectors: Now sum up the field vectors: EA EB EC E = + + = 22500i + 11250cos45i + 11250sin45j - 22500j = 30455i - 14545j (n/C)

14. Electric field far from a dipole FYI: The dipole electric field drops off by 1/x3 rather than inverse square like a "monopole." Topic 6.2 ExtendedB – Electric field What is the electric field along the bisector of a distant dipole having a charge separation d? A dipole is an equal positive and negative charge in close proximity to one another: +q r d 2   d 2 x r -q kq r2 |E| = Both field vectors have the same length: Note that the horizontal components cancel, and the vertical components add: |ETOTAL| = 2|E|sin. But sin = (d/2)/r = d/2r so that kqd r3 |ETOTAL| = 2|E|sin = Since x >> d, r x and we have But r = (d/2)2 + x2. kqd x3 EDIPOLE=