# Not quite complete! Maxwell’s Equations We now have four formulas that describe how to get electric and magnetic fields from charges and currents Gauss’s.

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Not quite complete! Maxwell’s Equations 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!

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! II 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

Ampere’s Law (New Recipe) Is this self-consistent? Consider two surfaces with the same boundary B Gauss’s Law for electric fields: This makes sense! I1I1 I2I2  E2  E1

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

Gauss’s Law for Small Volumes (2) 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: a a a x yz Now include the other four faces:

Gauss’s Law for Small Volumes Divide both sides by a 3, 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: a a a

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 a a x y z Add the left and right sides Calculate the electric flux Put it together

Ampere’s Law for Small Loops (2) Divide by a 2 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 a x y z

Maxwell’s Equations: Differential Form In more sophisticated notation:

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