Prof. David R. Jackson ECE Dept. Fall 2014 Notes 18 ECE 2317 Applied Electricity and Magnetism 1

Example Find curl of E : s0s0 l0l0 q Infinite sheet of charge (side view) Infinite line charge Point charge 2

Example (cont.) s0s0 1 x 3

l0l0 2 4

q 3 5

This gives us Faraday’s law: (in statics) 6 Note: If the curl of the electric field is zero for the field from a point charge, then by superposition it must be zero for the field from any charge density.

Here S is any surface that is attached to C. Stokes's theorem: 7 Faraday’s Law in Statics (Integral Form) (Integral Form) Hence C

Faraday’s Law in Statics (Differential Form) (Differential Form) Hence We then have: We show here how the integral form also implies the differential form. 8 Assume

Faraday’s Law in Statics (Summary) Integral form of Faraday’s law Differential (point) form of Faraday’s law Stokes’s theorem Definition of curl 9

Path Independence and Faraday’s Law The integral form of Faraday’s law is equivalent to path independence of the voltage drop calculation. 10 Proof Hence A B C C1C1 C2C2

Summary of Path Independence Equivalent Equivalent properties of an electrostatic field 11

Summary of Electrostatics 12 Here is a summary of the important equations related to the electric field in statics. Electric Gauss law Faraday’s law Constitutive equation

Faraday’s Law: Dynamics Experimental Law (dynamics): 13 This is the general Faraday’s law in dynamics. Michael Faraday* *Ernest Rutherford stated: "When we consider the magnitude and extent of his discoveries and their influence on the progress of science and of industry, there is no honour too great to pay to the memory of Faraday, one of the greatest scientific discoverers of all time". (from Wikipedia)

Magnetic field B z (increasing with time) Electric field E A ssume a B z that increases with time. Faraday’s Law: Dynamics (cont.) x y Experiment 14 The changing magnetic field produces an electric field.

Faraday’s Law: Integral Form Apply Stokes’ theorem for the LHS: 15 Integrate both sides over an arbitrary open surface (bowl) S : Faraday's law in integral form Note: The right-hand rule determines the direction of the unit normal, from the direction along C.

Faraday’s Law (Experimental Setup) Magnetic field B ( B z is increasing with time) Note: The voltage drop along the wire is zero. x y + - v(t) > 0 We will measure a voltage across a loop due to a changing magnetic field inside the loop. 16 Open-circuited loop

Faraday’s Law (Experimental Setup) Note: The voltage drop along the PEC wire is zero. So we have Also 17 x y + - S C A B v(t) > 0

Faraday’s Law (Experimental Setup) x y + - v(t) > 0 A = area of loop C A B Assume Assume a uniform magnetic field for simplicity (at least uniform over the loop area). 18 so

Faraday’s Law (Flux Form) x y + - v(t) > 0 Assume Assume a stationary loop (not changing with time). 19 Define  = magnetic flux through loop in z direction Then

Lenz’s Law A = area of loop Assume 20 B z is increasing with time. This is a simple rule to tell us the polarity of the output voltage. The voltage polarity is such that it would correspond to a current flow that would oppose the change in flux in the loop (if there were a resistor connected to the output so that current is allowed to flow). Note: A right-hand rule tells us the direction of the magnetic field due to a wire carrying a current. (A wire carrying a current in the z direction produces a magnetic field in the positive  direction.) x y + - v(t) > 0 I A B R

Example: Magnetic Field Probe Assume A small loop can be used to measure the magnetic field (for AC). 21 Assume Then we have At a given frequency, the output voltage is proportional to the strength of the magnetic field. x y + - v(t)v(t) A = area of loop C

Applications of Faraday’s Law 22 Faraday’s law explains:  How electric generators work  How transformers work Output voltage of generator Output voltage on secondary of transformer Note: For N turns in the loop we have

The world's first electric generator! (invented by Michael Faraday) 23 A magnet is slid in and out of the coil, resulting in a voltage output. (Faraday Museum, London)

The world's first transformer! (invented by Michael Faraday) 24 The primary and secondary coils are wound together on an iron core. (Faraday Museum, London)

AC Generators 25 Diagram of a simple alternator (AC generator) with a rotating magnetic core (rotor) and stationary wire (stator), also showing the current induced in the stator by the rotating magnetic field of the rotor. Note: If the magnetic field from the magnet is constant but the magnetic rotates at a fixed speed, a sinusoidal voltage output is produced. z + - A = area of loop so or

26 Generators at Hoover Dam AC Generators (cont.)

27 Generators at Hoover Dam AC Generators (cont.)

28Transformers A transformer changes an AC signal from one voltage to another.

29Transformers  High voltages are used for transmitting power over long distances (less current means less conductor loss).  Low voltages are used inside homes for convenience and safety.

30 Transformers (cont.) Ideal transformer A = area of core (phasor domain) Hence (time domain)

31 Transformers (cont.) Ideal transformer (no losses): Hence (phasor domain) so (time domain)

32 Transformers (cont.) Impedance transformation (phasor domain): Hence

Maxwell’s Equations (Differential Form) Electric Gauss law Magnetic Gauss law Faraday’s law Ampere’s law Constitutive equations 33

Maxwell’s Equations (Integral Form) Electric Gauss law Magnetic Gauss law Faraday’s law Ampere’s law 34

Maxwell’s Equations (Statics) Electrostatics Magnetostatics In statics, Maxwell's equations decouple into two independent sets. 35

Maxwell’s Equations (Dynamics) In dynamics, the electric and magnetic fields are coupled together. Example: A plane wave propagating through free space E H power flow z From ECE 3317: 36 Each one, changing with time, produces the other one.