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  • Today I'm going to work with you on a new concept and that is

  • the concept of what we call electric field.

  • We spend the whole lecture on electric fields.

  • If I have a -- a charge, I just choose Q,

  • capital Q and plus at a particular location and at

  • another location I have another charge little Q,

  • I think of that as my test charge.

  • And there is a separation between the two which is R.

  • The unit vector from capital Q to li- little Q is this vector.

  • And so now I know that the two charges if they were positive --

  • let's suppose that little Q is positive, they would repel each

  • other. Little Q is negative they would

  • attract each other. And let this force be F and

  • last time we introduced Coulomb's law that force equals

  • little Q times capital Q times Coulomb's constant divided by R

  • squared in the direction of R roof.

  • The two have the same sign. It's in this direction.

  • If they have opposite sign it's in the other direction.

  • And now I introduce the idea of electric field for which we

  • write the symbol capital E.

  • And capital E at that location P where I have my test charge

  • little Q at that location P is simply the force that a test

  • charge experienced divided by that test charge.

  • So I eliminate the test charge. So I get something that looks

  • quite similar but it doesn't have the little Q in it anymore.

  • And it is also a vector. And by convention,

  • we choose the force such that if this is a positive test

  • charge then we say the E field is away from Q if Q is positive,

  • if Q is negative the force is in the other direction,

  • and therefore E is in the other direction.

  • So we adopt the convention that the E field is always in the

  • direction that the force is on a positive test charge.

  • What you have gained now is that you

  • have taken out the little Q. In other words the force here

  • depends on little Q. Electric field does not.

  • The electric field is a representation for what happens

  • around the charge plus Q. This could be a very

  • complicated charge configuration.

  • An electric field tells you something about that charge

  • configuration. The unit for electric field you

  • can see is newtons divided by coulombs.

  • In SI units and normally we won't even indicate the um the

  • unit, we just leave that as it is.

  • Now we have graphical representations for the electric

  • field. Electric field is a vector.

  • So you expect arrows and I have here an example of a -- a charge

  • plus three. So by convention the arrows are

  • pointing away from the charge in the same direction that a

  • positive test charge would experience the

  • force. And you notice that very close

  • to the charge the arrows are larger than farther away.

  • That it that sort of represents is trying to represent the

  • inverse R square relationship. Of course it cannot be very

  • quantitative. But the basic idea is this is

  • of course spherically symmetric, if this is a point charge.

  • The basic idea is here you see the field vectors and the

  • direction of the arrow tells you in which direction the force

  • would be. If it is a positive test

  • charge. And the length of the vector

  • give you an idea of the magnitude.

  • And here I have another charge minus one.

  • Doesn't matter whether it is minus one coulomb or minus

  • microcoulomb. Just it's a relative

  • representation. And you see now that the E

  • field vectors are reversed in direction.

  • They're pointing towards the minus charge by convention.

  • And when you go further out they are smaller and you have to

  • go all the way to infinity of course

  • for the field to become zero. Because the one over R square

  • field falls off and you have to be infinitely far away for you

  • to not experience at least in principle any effect from the

  • from the charge. What do we do now when we have

  • more than one charge? Well, if we have several

  • charges -- here we have Q one and here we have Q two and here

  • we have Q three and let's say here we

  • have Q of I, we have I charges.

  • And now we want to know what is the electric field at point P.

  • So it's independent of the test charge that I put here.

  • You can think of it if you want to as the the force per unit

  • charge. You've divided out the charge.

  • So now I can say what is the E field due to Q one alone?

  • Well, that would be if Q one were

  • positive then this might be a representation for E one.

  • If Q two were negative, this might be a representation

  • for E two, pointing towards the negative charge.

  • And if this one were negative, then I would have here a

  • contribution E three, and so on.

  • And now we use the superposition principle as we

  • did last time with Coulomb's law, that the

  • net electric field at point P as a vector is E one in

  • reference of charge Q one plus the vector E two plus E three

  • and so on and if you have I charges it is the sum of all I

  • charges of the individual E vectors.

  • Is it obvious that the superposition principle works?

  • No. Does it work?

  • Yes. How do we

  • know it works? Because it's consistent with

  • all our experimental results. So we take the superposition

  • principle for granted and that is acceptable.

  • But it's not obvious. If you tell me what the

  • electric field at this point is which is the vectorial sum of

  • the individual E field vectors then I can always tell you what

  • the force will be if I bring a charge at that location.

  • I take any charge that I always would carry in my pocket,

  • I take it out of my pocket and I put it at that location.

  • And the charge that I have in my pocket is little Q.

  • Then the force on that charge is always Q times E.

  • Doesn't matter whether Q is positive.

  • Then it will be in the same direction as E.

  • If it is negative it will be in the opposite direction as E.

  • If Q is large the force will be large.

  • If Q is small the force will be small.

  • So once you know the E field it could be the result of

  • very complicated charge configurations,

  • the real secret behind the concept of an E field is that

  • you bring any charge at that location and you know what force

  • acts at that point on that charge.

  • If we try to be a little bit more quantitative,

  • suppose I had here a charge plus three and here I had a

  • charge minus one. Here's minus one.

  • And I want to know what the field configuration is as a

  • result of these two charges. So you can go to any particular

  • point. You get an E vector which is

  • going away from the plus three, you get one that goes to minus

  • one, and you have to vectorially add the two.

  • If you are very close to minus one, it's very clear because of

  • the inverse R square relationship that the minus one

  • is probably going to win.

  • Let's in our mind take a plus test charge now.

  • And we put a plus test charge very close to minus one,

  • say put it here, even though plus three is

  • trying to push it out, clearly minus one is most

  • likely to win. And so there will probably be a

  • force on my test charge in this direction.

  • The net result of the effects of the two.

  • Suppose I take the same positive test charge and I put

  • it here, very far away, much farther away than this

  • separation. What do you think now is the

  • direction of the force on my plus charge?

  • Very far away. Excuse me.

  • Why do you think it's to the left?

  • Do you think minus one wins? A: [inaudible].

  • Do you really think the minus one is stronger than the plus

  • three because the plus three will push it out and

  • the minus one tries to lure it in, right, if the test charge is

  • positive. A:

  • plus two. So if you're far away from a

  • configuration like this, even if you were here,

  • or if you were there, or if you're way there,

  • clearly the field is like a plus two charge.

  • And falls off as one over R squared.

  • So therefore if you're far away the force is in this direction.

  • And now look, what is very interesting.

  • Here if you're close to the minus one, the force is in this

  • direction. Here when you're very far away,

  • maybe I should be all the way here, it's in that direction.

  • So that means there must be somewhere here

  • the point where the E field is zero.

  • Because if the force is here in this direction but ultimately

  • turns over in that direction, there must be somewhere a point

  • where E is zero. And that is part of your

  • assignment. I want you to find that point

  • for a particular charge configuration.

  • So let's now go to some graphical representations of a

  • situation which is actually plus three minus one,

  • try to improve on the light situation.

  • And let's see how these electric vectors,

  • how they show up in the vicinity of these two charges.

  • So here you see the plus three and the minus one,