Faraday’s law cannot be derived from the other fundamental principles we have studied Formal version of Faraday’s law: Sign: given by right hand rule Faraday’s Law Differential form of Faraday’s law:

‘Magnetic force’ approach: I Use Faraday law: I Faraday’s Law and Motional EMF

I Faraday’s Law and Generator

A uniform time-independent magnetic field B=3 T points 30 o to the normal of the rectangular loop. The loop moves at constant speed v 1. What is the emf? 2. In 0.1 s the loop is stretched to be 0.12 m by 0.22 m. What is average emf during this time? Exercise

B1B1 B2B2 v L R Lv  t I Example

Two ways to produce curly electric field: 1. Changing B 2. Changing A Two Ways to Produce Changing 

1.Loop is moving: motional emf 2.Coil is moving: changing B Reference Frame

Three kinds of electric and magnetic effects on electrons in a wire 1. Coulomb electric field due to surface charges 2. Time-varying magnetic field leads to curly electric fields 3. Magnetic forces if the wire is moving The round-trip integral of a Coulomb electric field is zero The round trip integral of a non-Coulomb electric field and non- Coulomb magnetic force per unit charge together gives emf (Faraday’s law) Non-Coulomb Fields and Forces

Faraday’s law: summarizes a wide variety of physical phenomena correctly predicts the observed electric field But it does not explain why. Physical law: can explain phenomena but does not tell why Similar laws: Gauss’s law, Coulomb’s law, Biot-Savart law, Ampere’s law…. Fundamental laws: Einstein theory of special relativity Quantum electrodynamics Ultimate goal: ‘Theory of everything’ but would it explain itself? The Character of Physical Laws

Resistivity versus temperature for an ordinary metal Resistivity versus temperature for ‘superconductor’ infinite mobility! Superconductors

Lead wire at 7.2°K: infinite mobility I Current can run forever even if E=0! How can we detect if there is current? Does it violate the principle of conservation of energy? P=RI 2 Does a permanent magnet violate the principle of conservation of energy? Infinite Mobility

First superconductor: 1911, Kamerling Onnes, mercury becomes superconductor at <4°K. Late 1980’s: New class of superconductors at ~77°K Importance: Energy losses in wires. Discovery

1.Cool it down 2.Move magnet What will happen? Infinite current is impossible!   mag cannot change. Current in the loop will produce its own B to compensate for any changes in magnetic flux. Magnetic Flux Through a Superconducting Ring

1.  magnet =constant, I=0 2.  magnet decreases I increases 3. Current creates  loop = -  magnet Why does it not happen in a regular metal wire? What will happen if we move the magnet back to its old location? Magnetic Flux Through a Superconducting Ring

What will happen if there is a solid disk instead of a loop? 1933: magnetic field is zero in type I superconductors (Meissner effect) Quantum-mechanical property Is there any force between the magnet and the disk? Levitation The Meissner Effect

Constant voltage – constant I, no curly electric field. Increase voltage: dB/dt is not zero  emf For long solenoid: Change current at rate dI/dt: (one loop) emf bat R emf coil Inductance

emf bat R emf coil ECEC Increasing I  increasing B E NC emf bat R emf ind L – inductance, or self-inductance Inductance

E NC ECEC emf bat R emf ind L Unit of inductance L: Henry = Volt. second/Ampere Inductance Increasing the current causes E NC to oppose this increase

ECEC E NC emf bat R emf ind L Conclusion: Inductance resists changes in current Inductance: Decrease Current Orientation of emf ind depends on sign of dI/dt

What is self-inductance of for a solenoid with 1000 loops wound on a rod 10 cm long and radius 1 cm? Example

Magnetic Field Energy Density? LI2I2

Electric and magnetic field energy density: Field Energy Density

If t is very long: Current in RL Circuit

If t is zero: Current in RL circuit: Current in RL Circuit

Current in RL circuit: Time constant: time in which exponential factor become 1/e Time Constant of an RL Circuit

22.P.30

a=0 Current in an LC Circuit

Current in an LC circuit Period: Frequency: Current in an LC Circuit

Non-ideal LC Circuit

Initial energy stored in a capacitor: At time t=0: Q=Q 0 At time t= : Q=0 System oscillates: energy is passed back and forth between electric and magnetic fields. Energy in an LC Circuit 1/4 of a period

What is maximum current? At time t=0: At time t= : Energy in an LC Circuit

Energy in LC Circuit (No dissipation in this circuit) As capacitor loses charge, current increases As capacitor gains charge, current decreases Same equation as obtained via considering potential differences

Frequency: Radio receiver: LC Circuit and Resonance

AC source Self induced emf opposes emf of an AC source making current smaller If number of loops is very large there will be almost no current in the circuit and emf ind will be equal to emf AC of the AC source: AC Current and a Coil

AC source AC Current and a Coil: Add a Loop

Energy conservation: Transformer

Varying B is created by AC current in a solenoid What is the current in this circuit? Advantage of using AC: Currents and emf ‘s behave as sine and cosine waves. Two Bulbs Near a Solenoid

Add a thick wire: Loop 1 Loop 2 I1I1 I2I2 I3I3 Loop 1: Loop 2: Node: Two Bulbs Near a Solenoid

Add a thick wire: Loop 1 Loop 2 I1I1 I2I2 I3I3 Loop 1: Loop 2: Node: Two Bulbs Near a Solenoid

Exercise

E NC ECEC Electric Field in a Non-uniform Ring

emf due to non-coulomb electric field What is the second term due to? Motional emf: Magnetic force! Changing Area and B Simultaneously

Exercise

A check on the direction of the curly, non-Coulomb electric field and the current flow (does not give magnitude!) The direction of the curly non-Coulomb electric field is such that the current it drives is in a direction that makes a magnetic field that attempts to keep the flux constant. Lenz’s Rule