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