Lecture 37: WED 22 APR CH32: Maxwell’s Equations I James Clerk Maxwell ( ) Physics 2113 Jonathan Dowling

Maxwell I: Gauss’ Law for E-Fields: Maxwell I: Gauss’ Law for E-Fields: charges produce electric fields, field lines start and end in charges S S SS

Maxwell II: Gauss’ law for B-Fields: Maxwell II: Gauss’ law for B-Fields: field lines are closed or, there are no magnetic monopoles S SS S SS

Maxwell III: Ampere’s law: Maxwell III: Ampere’s law: electric currents produce magnetic fields C

Maxwell IV: Faraday’s law: Maxwell IV: Faraday’s law: changing magnetic fields produce (“induce”) electric fields

Maxwell Equations I – IV:

In Empty Space with No Charge or Current …very suspicious… NO SYMMETRY! ? q=0 i=0

Maxwell’s Displacement Current If we are charging a capacitor, there is a current left and right of the capacitor. Thus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires. But no magnetic field inside the capacitor? With a compass, we can verify there is indeed a magnetic field, equal to the field elsewhere. But Maxwell reasoned this without any experiment! But there is no current producing it! ? E BB The missing Maxwell Equation!

Maxwell’s Fix We can write the current as: We calculate the magnetic field produced by the currents at left and at right using Ampere’s law :E i d =  0 d   /dt

B ! E i B i B Displacement “Current” Maxwell proposed it based on symmetry and math — no experiment! Changing E-field Gives Rise to B-Field!

Maxwell’s Equations I – V: I II III IV V

Maxwell Equations in Empty Space: Fields without sources? Changing E gives B. Changing B gives E. Positive Feedback Loop!

32.3: Induced Magnetic Fields: Here B is the magnetic field induced along a closed loop by the changing electric flux  E in the region encircled by that loop. Fig (a) A circular parallel-plate capacitor, shown in side view, is being charged by a constant current i. (b) A view from within the capacitor, looking toward the plate at the right in (a).The electric field is uniform, is directed into the page (toward the plate), and grows in magnitude as the charge on the capacitor increases. The magnetic field induced by this changing electric field is shown at four points on a circle with a radius r less than the plate radius R.

32.3: Induced Magnetic Fields: Ampere Maxwell Law: Here i enc is the current encircled by the closed loop. In a more complete form, When there is a current but no change in electric flux (such as with a wire carrying a constant current), the first term on the right side of the second equation is zero, and so it reduces to the first equation, Ampere’s law.

Example, Magnetic Field Induced by Changing Electric Field:

Example, Magnetic Field Induced by Changing Electric Field, cont.:

32.4: Displacement Current: Comparing the last two terms on the right side of the above equation shows that the term must have the dimension of a current. This product is usually treated as being a fictitious current called the displacement current i d : in which i d,enc is the displacement current that is encircled by the integration loop. The charge q on the plates of a parallel plate capacitor at any time is related to the magnitude E of the field between the plates at that time by in which A is the plate area. The associated magnetic field are: AND

Example, Treating a Changing Electric Field as a Displacement Current:

32.5: Maxwell’s Equations: