Clicker Question: Non-uniform B field A current loop is oriented so that its magnetic dipole is oriented along direction “2”. In which direction is the force on the loop? In direction 1 In direction 2 In direction 3 In direction 4 No force B, magnetic field increases in the direction of 2 so F= mu (dB/dx) . Gives a force towards stronger field strength

Clicker Question The plot shows I(t) for an R-C circuit where a fully charged capacitor is discharged through the resistor. What is the approximate value of the RC-time constant for this circuit? 3 s; 8 s; 15 s; 25 s; 30 s

Gauss’s Law Divergence theorem: Differential form of Gauss’s law Provides local relationship between charge and electric field E and  are at the same location at the same time; that avoids the problem of relativistic retardation! Divergence:

Maxwell’s Equations (incomplete) Gauss’s law for electricity Gauss’s law for magnetism Incomplete version of Faraday’s law Ampere’s law (Incomplete Ampere-Maxwell law) Last shown on 11/6. First two: integrals over a surface Second two: integrals along a path Incomplete: no time dependence

Is Ampere’s Law Relativistically Correct? Problem: If currents inside change, changes in B will be delayed. To avoid retardation problem: find a property of B at (x,y,z,t) relate it to current at (x,y,z,t) ‘del’ operator

Differential Form of Ampere’s Law J A Current passing through tiny area: Differential form of Ampere’s law: curl – the component in n direction is the limit Note: Ampere’s law is incomplete – we will consider relativistic case of integral form later different notation Note: it is a vector equation it is relativistically correct

Chapter 23 Faraday’s Law

Electric and Magnetic Fields Electric and magnetic fields are interconnected Moving charges make magnetic fields Magnetic fields affect moving charges Electric and magnetic fields depend on reference frame Change reference frame: magnetic field is converted into electric Changing reference frame converts magnetic into electric etc No matter how E was produced it acts the same F=qE, but in the case of magnetic nature – can get curly pattern, I.e. produce emf in a loop of wire Time varying magnetic field can produce electric field! (magnetic induction)

Changing Magnetic Field Solenoid: inside outside Constant current: there will be no forces on charges outside (B=0, E=0) What if current is not constant in time? Let B increase in time What will happen if we put a loop of wire in? Round trip is not zero! Got curly electric field E~dB/dt E~1/r Non-Coulomb ENC !

Two Ways to Produce Electric Field 1. Coulomb electric field: produced by charges 2. Non-Coulomb electric field: using changing magnetic field 𝐸 1 ~ 𝑑 𝐵 1 𝑑𝑡 ~ 1 𝑟 Field outside of solenoid Same effect on charges: What will happen if we put a loop of wire in?

Direction of the Curly Electric Field Right hand rule: Thumb in direction of fingers: ENC Exercise: Magnetic field points down from the ceiling and is increasing. What is the direction of E? Clockwise viewed from below.

Driving Current by Changing B ENC causes current to run along the ring What is the surface charge distribution? What is emf and I? What will happen if we put a loop of wire in? It can make a current run in a wire just as though a battery was present! Ring has resistance, R

Effect of the Ring Geometry 1. Change radius r2 by a factor of 2. emf does not depend on radius of the ring! Double radius -> half the electric field 2. One can easily show that emf will be the same for any circuit surrounding the solenoid

Round-Trip Not Encircling the Solenoid =0 =0 + _ for any path not Enclosing solenoid! The non-coulomb electric field will polarize the wire, but would not drive a current around the loop. Since integral does not depend on radial distance (due to cancellation of 1/r dependence of E_NC and r dependence of dl ( =r*d(theta) ), emf = 0.

Exercise Is there current in these circuits?

Quantitative Relationship Between B and EMF Can observe experimentally: I=emf/R ENC~emf Increase current -> current runs clockwise out if the + terminal of the ammeter emf does not depend on loop geometry! Our demo set is not good enough to detect this emf, need more sensitive ammeter etc ENC~dB/dt ENC~ cross-section of a solenoid

Magnetic Flux - magnetic flux mag on the area encircled by the circuit Magnetic flux on a small area A: Magi. flux is calc the same way as electric flux Definition of magnetic flux: This area does not enclose a volume!

Faraday’s Law Michael Faraday (1791 - 1867) Faraday’s law cannot be derived from the other fundamental principles we have studied Formal version of Faraday’s law: 𝐸 𝑁𝐶 ∙𝑑 𝑙 =− 𝑑 𝑑𝑡 𝐵 ∙ 𝑛 𝑑𝐴 Unlike motional emf – Faraday cannot be derived . A British physicist and chemist, he is best known for his discoveries of electro-magnetic rotation, electro-magnetic induction and the dynamo. Faraday's ideas about conservation of energy led him to believe that since an electric current could cause a magnetic field, a magnetic field should be able to produce an electric current. He demonstrated this principle of induction in 1831. Faraday expressed the electric current induced in the wire in terms of the number of lines of force that are cut by the wire. The principle of induction was a landmark in applied science, for it made possible the dynamo, or generator, which produces electricity by mechanical means Sign: given by right hand rule

Including Coulomb Electric Field 𝐸 𝑁𝐶 ∙𝑑 𝑙 =− 𝑑 𝑑𝑡 𝐵 ∙ 𝑛 𝑑𝐴 Can we use total E in Faraday’s law? Enc = curry non-coulomb electric field with nonzero path integral Unlike motional emf – Faraday cannot be derived =0 𝐸 ∙𝑑 𝑙 =− 𝑑 𝑑𝑡 𝐵 ∙ 𝑛 𝑑𝐴

A Circuit Surrounding a Solenoid 𝐸 𝑁𝐶 Example: B1 changes from 0.1 to 0.7 T in 0.2 seconds; area=3 cm2. 𝑒𝑚𝑓=− ∆ Φ 𝑚𝑎𝑔 Δ𝑡 =− 0.6T 3×1 0 −4 m 2 0.2s =−9×1 0 −4 V Application of Faraday’s law Circuit acts as battery. What is the ammeter reading? (resistance of ammeter+wire is 0.5) 𝐼=𝑒𝑚𝑓/𝑅=−1.8×1 0 −3 V − 𝑑 𝐵 𝑑𝑡 downwards

A Circuit Not Surrounding a Solenoid If we increase current through solenoid what will be ammeter reading? Application of Faraday’s law

Voltmeter Reading Problem – when measuring dV keep wires which go to Voltmeter so that they do not encircle area with changing B Voltmeter = ammeter with large resistance

The EMF for a Coil With Multiple Loops Each loop is subject to similar magnetic field  emf of loops in series: Series connection of many loops with the same emf

Moving Coils or Magnets Time varying B can be produced by moving coil: by moving magnet: by rotating magnet: (or coil) move coil – get changing B and curly E How to create a time-varying magnetic field?

Exercise 1. A bar magnet is moved toward a coil. What is the ammeter reading (+/-)? + 𝐵 𝐵 2. The bar magnet is moved away from the coil. What will ammeter read? _ move coil – get changing B and curly E Positive Negative Negative (decreases with time as in 2) 3. The bar magnet is rotated. What will ammeter read? _ − 𝑑 𝐵 𝑑𝑡 ?

Complication Two loops: one produces changing B1 1 2 1 2 Two loops: one produces changing B1 If I2 changes in time it creates additional emf in the first loop!

Faraday’s Law and Motional EMF ‘Magnetic force’ approach: I 𝐿 Use Faraday law: 𝑣∆𝑡 When a wire moves through magnetic field -> magnetic forces drive current through wire I 𝐿

Example B1 B2 v L R I Lvt Old problem

Faraday’s Law and Generator I Old problem

Exercise A uniform time-independent magnetic field B=3 T points 30o 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? Old problem

Two Ways to Produce Changing  Two ways to produce curly electric field: 1. Changing B 2. Changing A Changing B field or area (shape, orientation)

Reference Frame Loop is moving: motional emf Coil is moving: changing B