Copyright © 2012 Pearson Education Inc. Applications of Electromagnetism in the News New battery technology Summer, Lauren (2013, November 1) Silicon Valley in Race for Battery Breakthrough KQED Science. valley-in-race-for-battery-breakthrough/ KQED Science valley-in-race-for-battery-breakthrough/ Understanding Earth’s Magnetic Field Redd, N. (2013, October 9) Weird Shift of Earth's Magnetic Field Explained. Space.com.

Copyright © 2012 Pearson Education Inc. PowerPoint ® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Chapter 28 Sources of Magnetic Field

Copyright © 2012 Pearson Education Inc. Goals for Chapter 28 To determine the magnetic field produced by a moving charge To study the magnetic field of an element of a current-carrying conductor To calculate the magnetic field of a long, straight, current-carrying conductor

Copyright © 2012 Pearson Education Inc. Goals for Chapter 28 To study the magnetic force between current- carrying wires To determine the magnetic field of a circular loop To use Ampere’s Law to calculate magnetic fields

Copyright © 2012 Pearson Education Inc. Introduction What can we say about the magnetic field due to a solenoid? What actually creates magnetic fields? We will introduce Ampere’s law to calculate magnetic fields. Check out CERN. The Large Hadron Collider (LHC) YouTube. Accessed 10/29/12 from CERNTV. The Large Hadron Collider YouTube. Accessed 10/19/12 from re=related re=related Euronews. Large Hadron Collider: A racetrack for particles. (2011) YouTube. Accessed 10/28/12 from

Copyright © 2012 Pearson Education Inc. The magnetic field of a moving charge A moving charge generates a magnetic field at some point in space around it. Charge q, moving with speed v in a direction Point P in space some distance away q v

Copyright © 2012 Pearson Education Inc. The magnetic field of a moving charge The field direction at that point depends upon the directions of the velocity vector AND the relative location of the point in space. Charge q, moving with speed v in a direction Point P in space some distance away q v B(r) r

Copyright © 2012 Pearson Education Inc. The magnetic field of a moving charge The field strength at that point depends on the speed and amount of the charge, the distance to the point, and the angle between v and r. Charge q, moving with speed v in a direction Point P in space some distance away q v B(r) r B = (  0 /4  q(v sin  )/r 2 

Copyright © 2012 Pearson Education Inc. The magnetic field of a moving charge A moving charge generates a magnetic field that depends on the velocity of the charge.

Copyright © 2012 Pearson Education Inc. The magnetic field of a moving charge A moving charge generates a magnetic field that depends on the velocity of the charge. B = (  0 /4  q(v x r)/r 3 = (  0 /4  q(v x r) r 2

Copyright © 2012 Pearson Education Inc. Magnetic force between moving protons What is  0 ? Magnetic “Permeability” of free space Like a magnetic dielectric? Related to electric permittivity of free space  0 Remember: B fields ~ PermeaBility E field ~ Voltage Difference ~ PermittiVity

Copyright © 2012 Pearson Education Inc. Magnetic force between moving protons What is  0 ?  0  0 = 1/c 2 Magnitude =  0 = 4  x Units = Tesla-meters/Amp (T-m/A) or Webers/Amp-meter (Wb/A-m)

Copyright © 2012 Pearson Education Inc. Ex 28.1 Magnetic force between moving protons E field exists moving or not! B Field of one moving particle creates FORCE on another moving particle

Copyright © 2012 Pearson Education Inc. Force on UPPER proton? E field direction? (+y) F (E field) = q 2 /4  0 r 2 F (B field) = qv x B (from lower particle) B field direction? (+z) B = (  0 /4  q(vi x j)/r 2 F B = (  0 /4  q 2 v 2 /r 2 F direction? (+y)

Copyright © 2012 Pearson Education Inc. Ratio of Forces on UPPER proton? F (B field) / F (E field)  0  0 v 2  = v 2 /c 2  At slow speeds, v <<c  F B <<F E

Copyright © 2012 Pearson Education Inc. Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. So a current of moving charges creates a B field!

Copyright © 2012 Pearson Education Inc. Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. So a current of moving charges creates a B field!

Copyright © 2012 Pearson Education Inc. Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. The law of Biot and Savart dB = (  0 /4  I(dl x r)/r 2

Copyright © 2012 Pearson Education Inc. Magnetic field of a current element The total magnetic field of several moving charges is the vector sum of each field. The law of Biot and Savart dB = (  0 /4  I(dl x r)/r 2

Copyright © 2012 Pearson Education Inc. Magnetic field of a straight current-carrying conductor Law of Biot & Savart to a long straight conductor: B =  0 I/2πx.

Copyright © 2012 Pearson Education Inc. Magnetic field of a current segment Example 28.2: What is B field of 125 Amp current from 1.0 cm segment of wire 1.2 meters away at two points?

Copyright © 2012 Pearson Education Inc. Magnetic field of a current segment Example 28.2: What is B field of 125 Amp current from 1.0 cm segment of wire 1.2 meters away at two points? For P 1 : B =  0 /4  I◦dl /r 2 (-z direction) For P 2 B =  0 /4  I ◦ dl ◦ sin (30) /r 2 (also -z direction) For P 1 : B =  x    z  For P 2 : B =  x    z 

Copyright © 2012 Pearson Education Inc. Magnetic fields of long wires Example 28.3 for one wire. Long, straight conductor carries 1.0 A; Where does B = 0.5 x T (about the equivalent of Earth’s field)? B = 0.5 x T =  0 I/2  r r =  0 I/2  B = 4 mm!

Copyright © 2012 Pearson Education Inc. Magnetic fields of long wires Example 28.4 for two wires. Find B at P 1, P 2, and P 3.

Copyright © 2012 Pearson Education Inc. Magnetic fields of long wires P 1 : -  0 I/2  d) +  0 I/2  (4d) = -  0 I/8  d P 2 : +  0 I/2  d) +  0 I/2  (d) = +  0 I/  d P 3 : +  0 I/2  d) -  0 I/2  (d) = -  0 I/3  d

Copyright © 2012 Pearson Education Inc. Force between parallel conductors The force per unit length on each conductor is F/L =  0 IIL/2πr. The conductors attract each other if the currents are in the same direction and repel if they are in opposite directions.

Copyright © 2012 Pearson Education Inc. Forces between parallel wires Example 28.5: What force does each wire exert on the other? F/L =  0 I I ’/2  r F/L = 1.0 x 10 4 N/m

Copyright © 2012 Pearson Education Inc. Magnetic field of a circular current loop The Biot Savart law gives B x =  0 Ia 2 /2(x 2 + a 2 ) 3/2 on the axis of the loop. At the center of N loops, where x = 0, the field on the axis is B x =  0 NI/2a.

Copyright © 2012 Pearson Education Inc. Magnetic field of a coil Figure (top) shows the direction of the field using the right-hand rule. Figure (below) shows a graph of the field along the x-axis.

Copyright © 2012 Pearson Education Inc. Ampere’s law (special case) Ampere’s law for a circular path around a long straight conductor.

Copyright © 2012 Pearson Education Inc. Ampere’s law (general statement) General statement of Ampere’s law

Copyright © 2012 Pearson Education Inc. Ampere’s law (general statement) General statement of Ampere’s law

Copyright © 2012 Pearson Education Inc. Ampere’s law (general statement) General statement of Ampere’s law

Copyright © 2012 Pearson Education Inc. Magnetic fields of long conductors Example 28.7 for a long straight conductor. Example 28.8 for a long cylinder.

Copyright © 2012 Pearson Education Inc. Field of a solenoid A solenoid consists of a helical winding of wire on a cylinder. Follow Example 28.9 using Figures 28.22–28.24 below.

Copyright © 2012 Pearson Education Inc. Field of a toroidal solenoid A toroidal solenoid is a doughnut-shaped solenoid. Follow Example using Figure below.

Copyright © 2012 Pearson Education Inc. The Bohr magneton and paramagnetism Follow the text discussions of the Bohr magneton and paramagnetism, using Figure below. Table 28.1 shows the magnetic susceptibilities of some materials. Follow Example

Copyright © 2012 Pearson Education Inc. Diamagnetism and ferromagnetism Follow the text discussion of diamagnetism and ferromagnetism. Figure at the right shows how magnetic domains react to an applied magnetic field. Figure below shows a magnetization curve for a ferromagnetic material.

Copyright © 2012 Pearson Education Inc. Hysteresis Read the text discussion of hysteresis using Figure below. Follow Example