Chapter 33 Maxwell’S Equations

Chapter 33 Maxwell’s Equations

q Gauss's law electric :  EAd S εo Gauss's law in (magnetic):  BAd 0 S d Faraday's law: Esd   B  dt d Ampere-Maxwell law Bsd μμI  E  oodt Faraday’s Law of Induction d  = -N B dt The emf induced in a closed loop is directly proportional to the time rate of change of the magnetic flux through the loop. The electric field is path dependent – E is NOT conservative and that is why we don’t call it a B POTENTIAL DIFFERENCE. Remember: V V   V  Es  d BA A for a conservative electric force and field. For an emf induced by a changing mag flux:

d d  E  ds  B   B  dA closeddt dt open Faraday’s Law of Induction The emf induced in a closed loop is directly proportional to the time rate of change of the magnetic flux through the loop. In this case the magnetic field in increasing in time which induces the electric field that creates the emf. Really, the changing magnetic field creates a nonconservative electric field which then produces the emf: dd  E  ds  B   B  dA closeddt dt open

The electric field is produced even if there is NO LOOP there. This is the key to understanding EM Waves!!! Michael Faraday

• 1791 – 1867 • British physicist and chemist • Great experimental scientist • Contributions to early electricity include: – Invention of motor, generator, and transformer – Electromagnetic induction – Laws of electrolysis Faraday’s Law of Induction

The emf induced in a closed loop is directly proportional to the time rate of change of the magnetic flux through the loop:

d  = -N B dt

- + Magnetic Flux • The magnetic flux associated with a magnetic field is defined in a way similar to electric flux:  BA d B  • The unit of magnetic flux is T.m2 = Wb is a weber • Gauss’ law in magnetism says  B d A = 0 the magnetic flux through any B  closed surface is always zero: NO MAGNETIC MONOPOLES

• In Faraday’s Law the surface is not closed so there can be a let flux usually through a LOOP.

d  = -N dt FLUX Magnetic FLUX The amount of FLOW perpendicular to a surface!  dd  BA  d BdAcos d d() BAcos  dt dt

The flux depends on the magnitude of the B field, the area, A, it intersects and the angle between them. Changes in B, A or  produce a change in the magnetic flux. d() BAcos   dt Magnetic Flux: Changing Angle

Magnetic Flux: B BA cos

Maximum Minimum Maximum Flux Flux Flux  = 0  = 90  = 180 Different forms of Faraday’s Law

dd  E  ds  B   B  dA closeddt dt open

d  = -N B dt

d() BAcos (AB cos )    =-N dt t Faraday’s Law How quickly you change the number of external field number of lines through one loop. loops

dext induced N Opposes the change dt in magnetic flux

The flux depends on the magnitude of the B field, the area, A, it intersects and the angle between them. Changes in B, A or  produce a change in the magnetic flux.

d() BAcos   dt Angle Changes Flux B Changes Flux Area Changes Flux Angle Changes Flux

d d( AB cos )  =-NN =- dt dt B Changes Flux d d( AB cos )  =-NN =- dt dt Area Changes Flux d d( AB cos )  =-NN =- dt dt EM Induction: Lenz’s Law How quickly you change the number of external field number of lines through one loop. loops

dext induced N Opposes the change dt in magnetic flux

Emf that drives the Lenz’s law: the induced current in a induced current loop is in the direction that creates a (nonconservative) magnetic field that opposes the change in magnetic flux through the area enclosed by the loop EM Induction: Lenz’s Law The induced EMF produces a a current that produces a magnetic field that opposes the original changing magnetic field.

To find the direction of the induced current: 1. Determine the flux change: Increasing? Decreasing? 2. The induced B field will be opposite. Find that. 3. Find the current with the RHR to produce the induced B field. Why does the Ring Jump?

Lenz’s Law! Changing B field The magnet is pushed into the coil. What is the direction of the induced current?

Lenz’s law: the induced - + current in a loop is in the direction that creates a B magnetic field that opposes the change in magnetic flux through the area enclosed by the loop Area Changes: Motional EMF d d( AB cos )  =-NN =- dt dt

What is the direction of the induced current in the circuit?

Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop A current-carrying wire is pulled away from a conducting loop in the direction shown. As the wire is moving, is there a cw current around the loop, a ccw current or no current?

1. There is a clockwise current around the loop. 2. There is a counterclockwise current around the loop. 3. There is no current around the loop.

Lenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop Area of Coil Changes

(AB cos )  =-N t (AB cos )  =-N t Area Changes: Motional EMF d d( AB cos )  =-NN =- dt dt • A motional emf is the emf induced in a conductor moving through a constant magnetic field. The Hall Effect induces an emf. • The electrons in the conductor experience a force, Fq v B that is directed along ℓ F qvB qE • The charges accumulate at vB E both ends of the conductor until they are in equilibrium with regard to the electric   V=El  Blv and magnetic forces. Sliding Conducting Bar Circuit

 = Blv

ε Bv I  RR

• A bar moving through a uniform field and the equivalent circuit diagram • Assume the bar has zero resistance and the stationary part of the circuit has a resistance R • The induced current obeys Ohms Law Forces on a Sliding Conducting Bar

FFapp B  vB FIBBB ()()  B RR vB22 FF app B R • The applied force to the right must balance the magnetic force to keep the bar moving at constant speed. The force does work on the bar. • The magnetic force on the induced current in the wire is opposite the motion of the bar. εMotional Emf  = Blv Geomagnetic Induction and Thrust

The forces generated by this "electrodynamic" tether can be used to pull or push a spacecraft to act as a brake or a booster or to generate voltage. Tethered Satellite Experiment

Calculate the motional emf induces alonga 20 km long conductor moving at the orbital speed of 7.8 km/s perpendicular to the Earth’s 0.5 microT magnetic field.  = Blv Cool Problem! Incline Generator!