Physics 2113 Jonathan Dowling Lecture 36: WED 19 NOV CH32: Maxwell s Equations II James Clerk Maxwell (1831-1879)
Maxwell s Displacement Current B E B 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? The missing Maxwell Equation! 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 Maxwell s Fix i d =ε 0 dφ E / We calculate the magnetic field produced by the currents at left and at right using Ampere s law : " C! B i d! s = µ 0 i ins i d = dq = d(cv ) We can write the displacement current as: = C dv = ε A 0 d d(ed) d(ea) = ε 0 q = VC C = ε 0 A / d V = Ed Φ E = dφ = ε E 0! " E i d A! = EA S
" C!" " d # B i ds = µ ε 0 0 C Displacement Current! B i d! s 0 B Maxwell proposed it based on symmetry and math no experiment! S B! " " E i d A B = µ 0 ε 0 dφ E i i E Changing E-field Gives Rise to B-Field!
32.3: Induced Magnetic Fields: Here B is the magnetic field induced along a closed loop by the changing electric flux F E in the region encircled by that loop. Fig. 32-5 (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.
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
32.4: Displacement Current:! " B i d s! dφ = µ 0 ε E 0 i d = ε 0 dφ E = µ 0 i d i d enc = i (r > R) i d enc = i πr 2 (r < R) 2 π R Using displacement current i d you can compute B without ever having to compute dφ E! i d enc i d!"
The displacement current i d = i is distributed evenly over grey area. So rank by i enc d = amount of grey area enclosed by each loop.! " B i d s! enc = µ 0 i d i d enc = ε 0 dφ E enc d = c > b > a
Example, Treating a Changing Electric Field as a Displacement Current: i d id
32.5: Maxwell s Equations:
The point where the field is perpendicular to Earth s surface and inward is not located at the geomagnetic north pole off Greenland as expected; instead, this so-called dip north pole is located in the Queen Elizabeth Islands in northern Canada, far from Greenland. 32.6: Magnets: The Magnetism of Earth: Because Earth s magnetic field is that of a magnetic dipole, a magnetic dipole moment µ is associated with the field. The field declination is the angle (left or right) between geographic north (which is toward 90 latitude) and the horizontal component of the field. The field inclination is the angle (up or down) between a horizontal plane and the field s direction. Magnetometers measure these angles and determine the field with much precision. One can do reasonably well with just a compass and a dip meter.
32.7: Magnetism and Electrons: Spin Magnetic Dipole Moment: An electron has an intrinsic angular momentum called its spin angular momentum (or just spin), S; associated with this spin is an intrinsic spin magnetic dipole moment, µ s. (By intrinsic, we mean that S and µ s are basic characteristics of an electron, like its mass and electric charge.) in which e is the elementary charge (1.60 x10-19 C) and m is the mass of an electron (9.11 10 31 kg).
32.7: Magnetism and Electrons: Spin Magnetic Dipole Moment: The orientation energy for the electron, when Bext is the exterior magnetic field aligned along the z-axis.
For an electron the spin is For a proton the spin is the opposite direction as the the same direction as the magnetic moment. magnetic moment. S! S! ( ) "! "! µ µ (a) Since (1) is uphill and (2) is downhill (2) is lower PE. S! ( ) "! µ S! "! µ S! ( ) "! µ S! "! µ (b) Since (1) is downhill and (2) is uphill (1) is lower PE. Uphill and downhill is with respect to µ!" not S ". Downhill in direction of B!".