Upcoming Schedule—FS 2003 “If you are out to describe the truth, leave elegance to the tailor.”—A. Einstein
F?? - - - - + + + The Electric Field Coulomb’s law (last lecture) presents us with several problems. Calculations become difficult as we increase the number of charges. Visualization becomes difficult.
http://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l4a.htmlhttp://www.glenbrook.k12.il.us/gbssci/phys/Class/estatics/u8l4a.html What do we do about continuous (approximately) distributions of charges? The Coulomb’s law force acts at a distance, without contact. How can we explain this?
The force of gravity acts at a distance, without contact. For a long time, this was difficult to for people to accept. We “invent” a gravitational field which “transmits” the gravitational force, and then we feel a lot better, because we feel like we’ve explained gravity. Click here for an animated view of the earth’s gravitational field. Now we have electrostatic forces acting at a distance, without contact.
To help us visualize how electric forces can act at a distance, we “invent” the electric field. The electric field “connects” the charges—we can draw it, and invoke it as the “thing” that provides “contact” for forces. Picture looks “funny” because I rotated it to give it an orientation corresponding to the top picture.
OSE:E = F / q An electric field extends outward from every + charge and permeates all of space. Other charges experience a force due to the electric field. Click the picture to see an animation. We define the electric field by specifying the force it exerts on an infinitesimally small* positive test charge q. *Why infinitesimally small? If it weren’t infinitesimally small, q itself would contribute to the electric field, exert forces on the other charges, possibly causing them to move, and in general foul up the measurement of electric field.
We can use our test charge to determine the electric field due to another charge Q located a distance r away. From Coulomb’s law, the magnitude of the force on q is so that the magnitude of the electric field due to Q is This is almost an OSE, but E is a vector, so we need to specify a direction.
Our test charge was positive, so the force exerted on it by another positive charge is repulsive. The electric field direction is the same as the force direction. The electric field points away from a positive charge. The principle of superposition applies to the electric field: the field due to a collection of charges is the vector sum of the fields due to the individual charges.
y A 30 cm 60 cm =30º x Q1=-50C Q2=+50C 52 cm Example 16-8: Calculate the total electric field at point A. Yes, this is almost the same setup as the example in the previous lecture.
The electric field is additive, so we can calculate EQ1 and EQ2 and add the two. Step 0: Think! This is an electric field problem (that’s what the problem statement says!). We want the electric field at point A due to charges Q1 and Q1. If we do our vector addition using components, we must resolve our electric fields into their x- and y-components.
EQ2 y EQ1 A 30 cm 60 cm =30º x Q1=-50C Q2=+50C 52 cm Step 1: Diagram Draw a representative sketch—done. r1 = Draw axes, showing origin and directions—done. r2= Draw and label relevant quantities—show the r’s. If there is no possibility of confusion, it is acceptable to write E1 instead of EQ1. If you do it in your diagram, do it in your equations too! Draw and label electric fields due to individual charges. Draw components of fields which are not along axes.
EQ2 r1 = y EQ1 A r2= 30 cm 60 cm =30º x Q1=-50C Q2=+50C 52 cm Step 2: OSE Step 3: Replace Generic Quantities by Specifics
EQ2 r1 = y EQ1 A r2= 30 cm 60 cm =30º x Q1=-50C Q2=+50C 52 cm Step 3 (continued) Remember, a vector has both magnitude and direction. A vector component contains direction information. Our convention is that subscripts x and y indicate vector components. The sign gives the direction if the component is along an axis, so a separate indication of direction is not needed.
Step 4: Complete the Math If there were two points, A and B, at which E was to be calculated, you would need to use the subscript A on the E’s here. I don’t use the subscript in this example because I am only doing the calculation at point A.
The boxed answers on the preceding slide are sufficient for a symbolic answer. Plugging in numbers gives Ex = 1.1x106 N/C and Ey = 4.4x106 N/C. It may look like my solution took more steps than necessary, but beware of skipped steps. Skipping steps is asking for trouble!
EQ2 r1 = y EQ1 A r2= 30 cm 60 cm =30º x Q1=-50C Q2=+50C 52 cm How could you calculate the force on an electron placed at point A (where we just calculated the electric field)? How would you calculate the electric field at point B, halfway between the two charges? B You could use symmetry arguments to calculate the electric field at any point halfway between the two charges. You are welcome to use symmetry arguments in your calculations. Just make sure it is valid to do so!
+ We could draw an arrow at every point in space. The direction would show the direction of E, and the length would show the magnitude. - Electric Field Lines Although the electric field helps us understand how electric forces can act at a distance, we may still have a visualization problem. Too many arrows! Too confusing!
A better approach is to use “field lines” to help us visualize the variation of the field in space. Field lines show the direction of the field, but not its magnitude. Field lines are also called “lines of force.” Field lines start at positive charges and end at negative charges. The electric field is tangent to the field lines. Make the number of lines proportional to the charge. The closer together the lines, the stronger the electric field.
The electric field in between two charged plates is (approximately) constant.
Electric Fields and Conductors If charges in a conductor at rest, the electric field inside must be zero. Why? If the electric field were not zero, the charges would experience a force and would accelerate. Any excess charge in a conductor spreads itself out on the surface. Why? Like charges repel. The charges move as far away from each other as possible.
A charge inside a spherical conducting shell gives rise to an electric field outside the shell, even though no field exists (i.e., the electric field is zero) inside the metal. Why? The electric field is always perpendicular to the surface outside of a conductor. Why? If there were a component of the electric field parallel to the surface, electrons would move along the surface until they came to a location where there was no parallel component.
Other Items of Interest E&M Fields. Little program that lets you explore fields and forces. An old dos program, runs fine in Windows XP. The fine print: click on the link, select “open,” and follow the prompts. This will copy a small (1.2 megabyte) set of files in the folder c:\Program Files\emfield. You can change the install folder if you want. After the files are copied, open windows explorer and run the file START.BAT in the emfield folder. To “uninstall” the emfield program, just delete the emfield folder and all the files in it. The program has been scanned for viruses, but you are always welcome to run your own scan. This force field applet also helps you explore fields and forces.
More Items of Interest Electric Field Hockey. Little program that lets you play hockey, using charges to move the “puck.” An old dos program, runs fine in Windows XP. The fine print: click on the link, select “open,” and follow the prompts. This will copy a small (0.8 megabyte) set of files in the folder c:\Program Files\hockey. You can change the install folder if you want. After the files are copied, open windows explorer and run the file START.BAT in the hockey folder. To “uninstall” the hockey program, just delete the hockey folder and all the files in it. The program has been scanned for viruses, but you are always welcome to run your own scan.
Even More Goodies! Some potentially useful links: The Physics Zone at Science Joy Wagon. Lessons on Electrostatics at the Physics Zone. JC Physics.com physics for A-level and Junior College Students (in other words, algebra-based physics).
If there were nothing more to electric fields than the material I have presented so far, they would not be much of a “big deal.” At best, they would give us a method slightly different than Coulomb’s law for calculating forces between electrical charges. At worst, they would confuse us by introducing the concept of “fields” that act on particles even though the fields aren’t made of anything we can touch. The abstraction I am about to make is very far-reaching.
Actually, we just use the vector dot product of the field with the area to find how much of the electric field is parallel to the surface normal: =EA, or d=EdA. Electric field lines extend outward from positive charges, towards negative charges, and the density of lines depends on the magnitude of the charge. We define electric flux as the number of electric field lines that pass perpendicularly through a surface.
The total flux through any surface is proportional to the charge inside the surface (remember, the number of electric field lines is proportional to the charge). Remember our constant k from Coulomb’s law? It is related to 0 by k=1/40. We could write using k, if we wish. But the total flux through a surface is the integral of d over the surface…
This gives us Gauss’ law: Gauss’ law is a BIG DEAL. It is one of Maxwell’s equations—the four fundamental equations that describe Electricity and Magnetism. You should think of Maxwell’s equations as being on the same fundamental level as our conservation laws. Sometimes you see physics majors walking around with t-shirts with a bunch of funny-looking equations on them. Maxwell’s equations! That little circle on the integral sign means integrate over a closed surface.
For the mathematicians in the audience, Gauss’ law can be written in an equivalent differential form. I’m not going to carry this any further here. If you took physics 24 or 25, you’d learn to love the integrals you’d do for Gauss’ law calculations. You can buy a Maxwell’s equation from the UMR Society of Physics Students and impress everybody with your knowledge of physics.
The Big Ideas Always go for the BIG IDEAS. These are “operational” big ideas; i.e., ones you need to understand to do well on tests. Big ideas you should have known before Physics 35: Superposition: Big concepts from this chapter: charge is conserved charges attract and repel
E = F / q Not really a “big idea” but something you must know: electric field lines start at + and end at - Official Starting Equations: Only 3 new OSE’s in the whole first chapter. This semester is going to be a piece of cake!