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Not quite complete!

Maxwell’s Equations. Not quite complete!. We now have four formulas that describe how to get electric and magnetic fields from charges and currents Gauss’s Law Gauss’s Law for Magnetism Ampere’s Law Faraday’s Law. There is also a formula for forces on charges Called Lorentz Force.

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Not quite complete!

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  1. Maxwell’s Equations Not quite complete! • We now have four formulas that describe how to get electric and magnetic fields from charges and currents • Gauss’s Law • Gauss’s Law for Magnetism • Ampere’s Law • Faraday’s Law • There is also a formula for forces on charges • Called Lorentz Force One of these is wrong!

  2. Ampere’s Law is Wrong! • Maxwell realized Ampere’s Law is not self-consistent • This isn’t an experimental argument, but a theoretical one • Consider a parallel plate capacitor getting charged by a wire • Consider an Ampere surface between the plates • Consider an Ampere surface in front of plates • But they must give the same answer! • There must be something else that creates B-fields • Note that for the first surface, there is also an electric field accumulating in capacitor • Maybe electric fields? • Take the time derivative of this formula • Speculate : This replaces I for first surface I I

  3. Ampere’s Law (New Recipe) I2 I1 E1 E2 • Is this self-consistent? • Consider two surfaces with the same boundary • Gauss’s Law for electric fields: B • This makes sense!

  4. Maxwell’s Equations • This is not the form in which Maxwell’s Equations are usually written • It involves complicated integrals • It involves long-range effects • Our first goal – rewrite them as local equations • Make the volumes very small • Make the loops very small • Large volumes and loops can be made from small ones • If it works on the small scale, it will work on the large Skip slides

  5. Gauss’s Law for Small Volumes (2) a a a z y x • Consider a cube of side a • One corner at point (x,y,z) • a will be assumed to be very small • Gauss’s Law says: • Let’s get flux on front and back face: • Now include the other four faces:

  6. Gauss’s Law for Small Volumes a a a • Divide both sides by a3, the volume • q/V is called charge density  • A similar computation works for Gauss’s Law for magnetic fields: • A more mathematically sophisticated notation allows you to write these more succinctly:

  7. Ampere’s Law for Small Loops • Consider a square loop a • One corner at point (x,y,z) • a will be assumed to be very small • Ampere’s Law says: • Let’s get integral on top and bottom z a y x a • Add the left and right sides • Calculate the electric flux • Put it together

  8. Ampere’s Law for Small Loops (2) z a y x • Divide by a2 • Current density J is I/A • Only in x-direction counts • Redo it for loops oriented in the other two directions • Similar formulas can be found for Faraday’s Law a

  9. Maxwell’s Equations: Differential Form • In more sophisticated notation:

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