Lecture 36: WED 18 NOV CH32: Maxwell s Equations I

Transcription

1 Physics 2113 Jonathan Dowling Lecture 36: WED 18 NOV H32: Maxwell s Equations I James lerk Maxwell ( )

2 Maxwell I: Gauss Law for E-Fields: charges produce electric fields, field lines start and end in charges " E d A = q / ε ins 0

3 Maxwell II: Gauss law for B- Fields: field lines are closed or, there are no magnetic monopoles " B i d A = 0

4 Φ = BA = 0 d > b > c > a = 0 T Φ &B a = = = 0 Φ ide T a = Φ &B a = 0 Φ ide a = 0 T Φ &B b = = 2 8 = 10 Φ ide T b = Φ &B b = 10 Φ ide b = 10 T Φ &B c = = = +4 Φ ide T c = Φ &B c = 4 Φ ide c = 4 T Φ &B d = = = +12 Φ ide T d = Φ &B d = 12 Φ ide d = 12

5 Maxwell III: Ampere s law: electric currents produce magnetic fields " B i d s = µ i 0 ins

6 Maxwell IV: Faraday s law: changing magnetic fields produce ( induce ) electric fields " E i d s = d B i d A = dφ B

7 Maxwell Equations I IV: " " E i d A = q / ε ins 0 B i d s = µ i 0 ins " B i d A = 0 " E i d s = d B i d A = dφ B

8 In Empty pace with No harge or urrent " " E i d A = 0 B i d A = 0? q=0 i=0 " B i d s = 0? " E i d s = d B i d A very suspicious NO YMMETRY = dφ B

9 Maxwell s Displacement urrent 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?

10 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 : " B i d s = µ 0 i ins i = dq = d(v ) We can write the current as: = dv = ε A 0 d d(ed) = ε 0 d(ea) q = V = ε 0 A / d V = Ed Φ E = dφ = ε E 0 " E i d A = EA

11 " " " d # B i ds = µ ε 0 0 Displacement urrent B i d s 0 B Maxwell proposed it based on symmetry and math no experiment B " " E i d A B = µ 0 ε 0 dφ E i i E hanging E-field Gives Rise to B-Field

12 Maxwell s Equations I V: I II IV E da = q / ε 0 B da = 0 d B ds = µ ε E da + µ 0 0 E ds = d V B da 0 i III

13 " " Maxwell Equations in Empty pace: E i d A = 0 B i d A = 0 hanging E gives B. hanging B gives E. Fields without sources? B i d s = + d " E i d A = + dφ E " E i d s = d B i d A = dφ B Positive Feedback Loop

14 32.3: Induced Magnetic Fields: Here B is the magnetic field induced along a closed loop by the changing electric flux Φ 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.

15 B a > B c > B b > B d = 0 dφ E = d EA = A de slope

16 The displacement current i d = i is distributed evenly over grey area. o 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

17 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.

18 Example, Magnetic Field Induced by hanging Electric Field:

19 Example, Magnetic Field Induced by hanging Electric Field, cont.:

20 32.4: Displacement urrent: omparing 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

21 Example, Treating a hanging Electric Field as a Displacement urrent:

22 Example, Treating a hanging Electric Field as a Displacement urrent: i d id

23 32.5: Maxwell s Equations: