Prof. David R. Jackson ECE Dept. Spring 2015 Notes 28 ECE 2317 Applied Electricity and Magnetism 1

Magnetic Field Lorentz force Law: In general, (with both E and B present): This experimental law defines B. The units of B are Webers/ m 2 or Tesla [ T ]. rr N N v q S N N S 2 B is the magnetic flux density vector. Note: Flux lines come out of north poles!

Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light is emitted along the electron path, due to the electrons colliding with gas molecules in the bulb. (From Wikipedia) Magnetic Field (cont.) 3 A stable orbit is a circular path. v                 F q > 0 q R x y

The most general stable path is a helix, with the helix axis aligned with the magnetic field. Magnetic Field (cont.) 4 There is no force in the axis direction (hence a constant velocity in this direction). Magnetic field lines thus “guide” charged particles.

The earth's magnetic field protects us from charged particles from the sun. Magnetic Field (cont.) 5 The particles spiral along the directions of the magnetic field, and are thus directed towards the poles.

This also explains the auroras seen near the north pole (aurora borealis) and the south pole (aurora australis). Magnetic Field (cont.) 6 The particles from the sun that reach the earth are directed towards the poles.

Magnetic Gauss Law x y z B S (closed surface) N Magnetic pole (not possible) ! No net magnetic flux out ! 7 Note: Magnetic flux lines come out of a north pole and go into a south pole. S N S

Magnetic Gauss Law: Differential Form From the definition of divergence we then have Hence 8

Ampere’s Law (This is an exact value: please see next slide.) Experimental law: 9 Note: A right-hand rule predicts the direction of the magnetic field. y x Iron filings I z

Ampere’s Law (cont.) Note: The definition of the Amp* is as follows: 1 [ A ] current produces: Hence so 10 *This is one form of the definition. Another form involves the force between two infinite wires.

Ampere’s Law (cont.) Define: The units of H are [ A/m ]. H is called the “magnetic field” (for single infinite wire) Hence 11

Ampere’s Law (cont.) y x I C A current (wire) is inside a closed path. 12 We integrate counterclockwise.

y x I C A current (wire) is outside a closed path. Ampere’s Law (cont.) 13 Note that the angle  smoothly goes from zero back to zero as we go around the path, starting on the x axis.

Hence I encl C “Right-Hand Rule” Although the law was “derived” for an infinite wire of current, the assumption is made that it holds for any shape current. This is now an experimental law. Ampere’s Law (DC currents) Ampere’s Law (cont.) Right-hand rule for Ampere’s law: The fingers of the right hand are in the direction of the path C, and the thumb gives the reference direction for the current that is enclosed by the path. (The contour C goes counterclockwise if the reference direction for current is pointing up.) 14 Note: The same DC current I encl goes through any surface that is attached to C.

Amperes’ Law: Differential Form C SS (from Stokes’s theorem)  S is a small planar surface. Since the unit normal is arbitrary (it could be any of the three unit vectors), we have 15

Maxwell’s Equations (Statics) Electric Gauss law Magnetic Gauss law Faraday’s law Ampere’s law 16

Maxwell’s Equations (Dynamics) Electric Gauss law Magnetic Gauss law Faraday’s law Ampere’s law 17

Displacement Current Ampere’s law: “Displacement current” (This term was added by Maxwell.) A current density vector J exists inside the lower plate. The charge Q on the lower plate increases with time. Capacitor being charged with DC current Motivation: 18 h A Insulator I z Q J

Ampere’s Law: Finding H 1) The “Amperian path” C must be a closed path. 2) The sign of I encl is from the right-hand rule. 3) Pick C in the direction of H (to the extent possible). Rules: 19

Example Calculate H An infinite line current along the z axis First solve for H . 20 x “Amperian path” y I C  z r

Example (cont.) 21 “Amperian path” y I C  z r

Example (cont.) 3 ) H  = 0 Magnetic Gauss law: 2 ) H z = 0 I C h  H z = 0 at  H  d  cancels  =  I S h 22

Example (cont.) y x I  z r Note: There is a “right-hand rule” for predicting the direction of the magnetic field. (The thumb is in the direction of the current, and the fingers are in the direction of the field.) 23

Example Coaxial cable  < a b <  < cb <  < c This inner wire is solid. The outer shield (jacket) of the coax has a thickness of t = c – b. Note: The permittivity of the material inside the coax does not matter here. C c a b x y r  b I I a z c 24 DC current Note: At DC, the current density is uniform, since the electric field is uniform (due to the fact that the voltage drop is path independent).

Example (cont.) This formula holds for any radius, as long as we get I encl correct. The other components are zero, as in the wire example. C c a b x y r  25

Example (cont.)  < a < a a <  < b b <  < c  > c Note: There is no magnetic field outside of the coax (a perfect “shielding property”). C c a b x y r  26

Example (cont.)  < a < a a <  < b b <  < c  > c Summary c a b x y  27

Example Solenoid z n = # turns/meter I a Calculate H 28 Ideal solenoid: n    < a < a Find H z h C  H  d  cancels H z is zero at 

Example (cont.)  > a > a Hence 29 h HzHz C  H z = 0 at  z H  d  cancels so

Example (cont.) The other components of the magnetic field are zero: 2) H  = 0 from 1) H  = 0 since C S 30

Example (cont.) Summary z n = # turns/meter I a 31 Note: A larger relative permeability will give a larger magnetic flux density. This results is a stronger magnet, or a larger inductance. Note: A right-hand rule can be used to determine the direction of the magnetic field (same rule as for a straight wire).

Example y x z Infinite sheet of current Calculate H y x Top view + side - side 32

Example (cont.) Hx-Hx- Hx+Hx+ y x (superposition with line currents) (magnetic Gauss Law) Also, by symmetry: 33

Note: There is no contribution from the left and right edges (the edges are perpendicular to the field). Example (cont.) w - side y x + side C 34

Note: The magnetic field does not depend on y. Example (cont.) Note: We can use a right hand-rule to quickly determine the direction of the magnetic field: Put your thumb is in the direction of the current, and your fingers will give the overall direction of the magnetic field. 35

Example h w I I x y z Find H everywhere Assume w >> h Parallel-plate transmission line 36 (We neglect fringing here.)

Example (cont.) y x h Two parallel sheets (plates) of opposite surface current 37

Example (cont.) 38 Magnetic field due to bottom plate Magnetic field due to top plate y x “bot” “top” h

Example (cont.) We then have Hence Recall that 39

Example (cont.) 40 We could also apply Ampere’s law directly: Hence Note: It is convenient to put one side (top) of the Amperian path in the region where the magnetic field is zero. y x h C xx

Low Frequency Calculations 41 At low frequency, the DC formulas should be accurate, as long as we account for the time variation in the results. x y i(t)i(t)  z r Example (line current in time domain)

Low Frequency Calculations (cont.) 42 We can also used the DC formulas in the phasor domain. x y i(t)i(t)  z r Example (line current in the phasor domain) (phasor H field)

Low Frequency Calculations (cont.) 43 Example (cont.) Converting from phasor domain to the time domain: (phasor H field)