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# week4 - 1 Monday October 10 Potentials I've already...

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1 Monday, October 10: Potentials I’ve already remarked on the similarity between the electrostatic force be- tween two point charges, F = q 1 q 2 r 2 , (1) and the gravitational force between two point masses, F = - G m 1 m 2 r 2 . (2) The similarities between the two suggest that we can profitably describe electrostatic phenomena using some techniques stolen (excuse me, I mean “borrowed”) from Newtonian gravitational theory. For instance, in dealing with gravity, it is useful to use the scalar field known as the gravitational potential. The gravitational potential Φ grav ( ~ r ) is defined such that the force ~ F grav acting on a unit mass is ~ F grav = - ~ Φ grav . (3) For an arbitrary mass density ρ ( ~ r ), the gravitational potential is Φ grav ( ~ r ) = - G Z ρ ( ~ r 0 ) | ~ r 0 - ~ r | d 3 r 0 . (4) Thus, you find the gravitational potential by smoothing the mass density field with a very broad (1 /r ) smoothing function. Minima in the potential correspond to dense regions in the universe. The gravitational potential itself cannot be directly measured. We can detect its gradient at a given point by determining the force on a unit mass. We can detect the difference in potential between two points by determining the energy needed to lift a particle from the point of lower potential to the point of higher potential. However, you can always add an arbitrary constant to the gravitational potential, Φ grav Φ grav + C , (5) without changing any of the physics. For a mass distribution of finite extent, it is conventional to choose C so that Φ grav 0 as ~ r → ∞ ; however, this is just a convention. The concept of the “potential” has proved to be very useful for Newtonian gravity (largely because it’s easier to deal with the scalar field 1

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Φ grav than with the vector field ~ F ). It would be doubly useful if we could also express electromagnetic phenomena in terms of potentials. Alas, there are a few complicating factors. It is true that the electrostatic force between two point charges is directly analogous to the gravitational force between two point masses. However, if the point charges are moving relative to each other, there is a magnetic force between them that has no analogy in Newtonian gravity. Thus, we need an extension of potential theory that will allow us to deal with magnetism. There’s another complicating factor, as well. Newtonian gravity only deals with slowly moving particles ( v ¿ c ). In studying electromagnetism, we must be prepared to deal with highly relativistic charged particles. With these caveats in mind, let’s see how we can define potentials for the electric field strength ~ E ( ~ r, t ) and the magnetic flux density ~ B ( ~ r, t ). Let’s start with the simplest of Maxwell’s equations: ~ ∇ · ~ B = 0 . (6) Because of the vector identity ~ ∇ · ( ~ ∇ × ~ f ) = 0 (7) for an arbitrary vector function ~ f , we see that the divergenceless magnetic flux density can be written in the form ~ B ( ~ r, t ) = ~ ∇ × ~ A ( ~ r, t ) , (8) where ~ A ( ~ r, t ) is the electromagnetic vector potential . 1 If we write ~ B as the curl of ~ A , we can write another of Maxwell’s equa- tions, ~ ∇ × ~ E = - 1 c ~ B
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