Examples of magnetic field calculations and applications. 1 Example of a magnetic moment calculation

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Examples of magnetic field calculations and applications. 1 Example of a magnetic moment calculation"

Transcription

1 Examples of magnetic field calculations and applications Lecture 12 1 Example of a magnetic moment calculation We consider the vector potential and magnetic field due to the magnetic moment created by a rotating surface charge, σ, on a cylinder. The geometry is shown in Figure 1. The magnetic moment, I(area), of a small loop at the position, z, as shown in the figure is ; d m = σ(rdθ dz) (πr 2 )ẑ = πσr 3 ω dz ẑ The vector potential is then; d A = µ 0 4π d m r r 3 = µ 0 4π [πσr3 ω ẑ r r 3 This is integrated using r = α z to get A. A = L/2 L/2 d A Choose α to lie in the (x, y) plane along the ˆx direction. Then ẑ r = αŷ A = µ 0 4π d m r r 3 = [ µ 0 4π [πσr3 ωα] ] dz L/2 L/2 dz 1 [α 2 + z 2 ] 3/2]ŷ The vector potential above has been calculated in the magnetic moment approximation, (ie in the first non-zero order expansion of the vector potential in a power series of the loop radius divided by the distance to the field point.) 2 The field and action of a Quadrupole lens The quadrupole field is illustrated by the magnetic field shown in Figure 2 and given by the equations; z = gx x = gz y = 0 1

2 P r z ω L/2 α dm R σ y x L/2 Figure 1: The vector potential of a rotating, cylindrical charge distribution z N S x S N Figure 2: A quadrupole magnet used to focus charged particles 2

3 The field is generated by wire windings that create the magnetic poles shown in the figure and parallel to the equipotential curves perpendicular to the field lines. We suppose the length of the field into the page is L, and the field lines are shown in the figure. The field strength increases linearly with distance from the axis. 2 = 2 x + 2 z = g 2 r 2 For positive particles moving with velocity V into the page, the Lorentz force converges a particle beam horizontally, and diverges it vertically. Check the force direction and note the further away from the axis, the stronger the force. A magnetic lens is created if two quadupole fields are placed in line and rotated by 90 deg with respect to each other. Then with a proper choice of spacing, and field strength can provide focusing of a parallel beam of particles to a point some distance behind the two magnets. 3 Power and the magnetic field The Lorentz force on a charge is F = q[ E + V ]. This force causes the charge to move in a direction perpendicular to the field and velocity, ˆl. Then we determine the power due to this movement. P = V F = q V E As indicated, the force term involving the magnetic field, V ( V ), vanishes, so the magnetic field does no work on a charge and cannot change its energy only the direction of its velocity. 4 Motion of a charged particle in a magnetic field We suppose a constant magnetic field in the ẑ direction, and a charged particle of mass, m, charge, q, and velocity, V moves in this field. The motion is given by the Lorentz force, which in non-relativistic form, is given by the kinetic equation, F = m a. m d V = q[ V ] Neglect the interaction of the charge with other charges which may be present in a beam of such particles. In Cartesian coordinates the coupled equation set below can be produced for = ẑ. 3

4 dv x dv y dv z = V y q m = V x q m = 0 The last equation requires that V z = constant. The first two equations may be decoupled giving a second order ode. d 2 V i 2 = q m V i In the above, V i represents V x or V y. The solution is harmonic, and to satisfy both first order equations; V x = V 0x cos(ωt + φ) V y = V 0x sin(ωt + φ) Where ω = q m and φ is a phase angle to be determined from the initial conditions. Assume that at t = 0 V x = V 0x and V y = 0. Then φ = 0 and A = V 0x. Now integrate these equations again to get the coordinate trajectories. Let the initial velocity in the ẑ direction be, V z = V 0z, and the initial position for (x, y, z) is, (0, (V 0x /ω), Z 0 ). z = V 0z t + Z 0 x = (V 0x /ω) sin(ωt) y = (V 0x /ω) cos(ωt) Substitution verifies these solutions and initial conditions. We observe the motion is a helix in 3-D with a projection of circular motion onto the (x, y) plane. The radius of this circle, R, is ; x 2 + y 2 = R 2 = [V 0x /ω] 2 In the above, R, represents the radius of curvature of the particle in the magnetic field. Insert the particle momentum projected onto the (x, y) plane, p. p = mv 0x = mωr = qr R = p q 4

5 5 Drift Velocity and the Lorentz force Suppose we apply a constant magnetic field in the ẑ direction and a constant electric field in the ˆx direction to a charged particle. The particle is initially at rest. The equations of motion in the (x, y) plane are; dv x dv y = q m V y + (q/m) E 0 = q m V x After application of the initial conditions, the solution has the form, with ω defined in the previous section; V x = (q/mω) sin(ωt) V y = (q/mω)[cos(ωt) E 0 /] The position is obtained by a second integration; x = (q/mω 2 )[cos(ωt)/ω] + X 0 y = (q/mω 2 )[sin(ωt)/ω (E 0 /)t] + Y 0 In the above (X 0, Y 0 ) is the initial position. This motion is strange as the circle center moves with constant velocity in the ŷ direction. Now we study this a little further by placing a resistance, proportional to the velocity, to the motion of the charge. This is artificial because force is proportional to acceleration not velocity, but you have used frictional forces proportional to velocity in mechanics, and here we want an energy disipating term. Thus add a term which contains the first odd derivative of the position, velocity. Introduction of the force term σ V into the above equations gives; dv x dv y + σ/m V x = q m V y + (q/m) E 0 + σ/m V y = q m V x Look for the equilibrum solution, ie the solution when the velocity becomes independent of time so that dv i = 0. V x = V y = qσ σ 2 + (q) 2 E 0 q 2 σ 2 + (q) 2 E 0 5

6 I V a dl b Figure 3: The geometry describing the Hall effect The drift angle between the applied field and the motion is called the Lorentz angle and is given by; tan(θ) = V y /V x = q/σ The particle drifts at a constant velocity at the angle θ with respect to the applied field. The above examples give a few simple illustrations of the motion of charged particles in magnetic fields. In general, this topic is treated in magnetohydrodynamics (plasma) for example, and the motion is highly non-linear and non-intuitive. The understanding of plasma is crucial to the development of controlled fusion reactors. 6 The Hall effect We suppose a current through a conducting medium in which there is a magnetic field perpendicular to the current flow and the surfaces of the conductor Figure 3. The current represents a flow of charge so that there is a deflection of the current due to the Lorentz force. As previously determined Idl = qv so the force on a small element of current due to the magnetic field when the magnetic field and the current are perpendicular is; df m = Idl Charge flows and builds up an electric field on the parallel surfaces of the conductor. When equilibrum is reached, the electric force cancels the magnetic force. F = 0 = q[ E + v ] Use in the above qv = Idl. When the force reaches equlibrium; 6

7 q(de) = Idl The charge per unit volume in the conductor as obtained from the figure is ρ = q/ab(dl). Substitute for q and let de = V/b, with V the potential between the sides of the conductor. Then; V = I/(ρa) The hall effect occurs for current flow in a magnetic field and is used in a number of instruments to measure either currents or magnetic fields. 7 Example of Ampere s law and superposition There is a current flow in a cylindrical conductor of radius, R. The conductor has a hole of radius, a, displaced a distance, b from the axis of the conductor. The geometry is shown in Figure 4. It is assumed that the current density is constant in the cylinder. We are to find the magnetic field in the hole. This is done by superposition of a current in a cylinder of radius a centered at z = b in opposition to the current in the cylinder, as shown in the figure. The current in the cylinder without the hole is; I T = JπR 2 The current in the hole flowing in the opposite direction is; I H = Jπa 2 Thus the total current in the cylinder with the hole is I T I a = I C I C = J(π[R 2 a 2 ]) J = I C π[r 2 a 2 ] Now find the field at point P in Figure 4 by superimposing the contributions to the field from each of these current densities. This can be done by Ampere s law. d l = µ 0 4π I For the conducting cylinder without the hole, the enclosed current is; I c = J c (πr 2 ). The circulation on the left side of the above integral is evaluated to be (2πr). 7

8 z ^ ^ φ φ z θ P r b θ r b a R Figure 4: The geometry to find the magnetic field inside a cylindrical cavity in a cylindrical conductor C = µ 0 I πr 2 π(r 2 a 2 ) 2πr = µ 0 Ir 2π(R 2 a 2 ) In the same way the field for the current creating the hole is; H = µ 0 Ir 2π(R 2 a 2 ) The geometry is shown in the figure. Subtract these vector fields. Write the unit vectors dzẑ = r d l and use the geometry to obtain z z = b. The final result is; = µ 0 Ib 2π(R 2 a 2 ) Thus the magnetic field is constant in the interior and pointed in the ẑ direction. 8 Magnetic pressure and energy Consider two parallel current sheets separated by a distance, d, with uniform, constant currents flowing in opposite directions, Figure 5. Find the magnetic field due to one of these sheets on the other. The magnetic field is obtained using Ampere s law as previously. ecause of symmetry the magnetic field must be directed parallel to the sheet, and can only depend on the perpendicular distance from the sheet to the field point. Evaluation of the integral form of Ampere s law gives; d l = 2L = µ0 I 8

9 L I/L d I/L Figure 5: The magnetic field created by 2 parallel current sheets with currents flowing in opposi te directions The factor of 2 comes from the field above and below the sheet, with L the distance parallel to the sheet of the path along. I is the current that flows through this Amperian loop. Thus for one sheet; = µ(i/l)/2 = (µ/2) I In the above we have written I as the current per unit wih on the sheet. The direction of the magnetic field is given by the right-hand-rule. Note that the field is independent of the distance from the sheet. Thus the fields when superimposed from the two sheets, add between the sheets and cancel outside the sheets. Finally we also see that the force generated by the magnetic field on one sheet interacting with the current on the other is repulsive. We visualize this situation by thinking of the magnetic field as creating a pressure between the plates tending to expand the distance between them. Use the Lorentz force to calculate this force. In the equation for the Lorentz force, substitute IL for qv. The field and current direction are perpendicular, so F 2 = I 2 L 1 = I 2 x 2 (µ/2)i 1. Now the total magnetic field between the plates comes from the superposition of both fields which add to T = 2(µ/2)I 1,2. Then I 1,2 = L/µ which we substitute for I 1 in the force equation yielding; F Lx = 2µ The above is the force per unit surface area (pressure) of one current sheet on the other. This pressure attempts to push the plates apart. Now suppose we do work against this pressure by compressing the plates a distance, d. The movment of the plates removes a volume of the magnetic field, Lxd, under the Amperian loop. and puts an energy into the system given by W = Fd. We remove the geometry in the equation by dividing by the volume to obtain the energy per unit volume which we assign to the magnetic field. If we allow the plates to 9

10 d l = µ0 I T I Figure 6: The geometry to find field enclosed in a torus using Ampere s law re-expand, the plates do work removing this energy creating additional field in this volume. Compare this energy density, (1/2µ 0 ) 2, to the energy density of the electric field, (ǫ 0 /2)E 2. 9 Additional Examples 9.1 Field of a torus The geometry of a torus is illustrated in Figure 6. We assume that a current sheet moving in a circular direction around the donught is continuous, (ie very tightly wound wires). No field leaks out of the donught, and the geometry is symmetric in azimuthal angle. Thus we can use Ampere s law to get the magnetic field. Use a circular Amperian loop as shown; = µ 0 2πr In this expression I T is the total current flowing. We could write a current per unit wih of I = I T 2πr to remove the geometric dependence. 10

11 z I dz r R1 x z R2 I y Figure 7: The geometry to find the vector potential created by two long, parallel filaments of current flowing in opposite directions 9.2 Field of a bipolar filament We assume two long filaments carring a current I in opposite directions. We are to find the vector potential, A, for this geometry, Figure 7. y symmetry, A, must be independent of z. We have; A = µ 0 4π dτ Jr The current density is in the direction of ẑ and use J d area. A = µ 0 4π ẑ [ dz (R2 2 + z2 ) 1/2 A = 2 µ 0 4π I ln[z + z 2 + R2 2 ] z + z 2 + R1 2 0 ẑ A = 2 µ 0 4π I ln[r 1 R 2 ] dz (R z2 ) 1/2] 11

Module 3 : Electromagnetism Lecture 13 : Magnetic Field

Module 3 : Electromagnetism Lecture 13 : Magnetic Field Module 3 : Electromagnetism Lecture 13 : Magnetic Field Objectives In this lecture you will learn the following Electric current is the source of magnetic field. When a charged particle is placed in an

More information

Handout 7: Magnetic force. Magnetic force on moving charge

Handout 7: Magnetic force. Magnetic force on moving charge 1 Handout 7: Magnetic force Magnetic force on moving charge A particle of charge q moving at velocity v in magnetic field B experiences a magnetic force F = qv B. The direction of the magnetic force is

More information

Chapter 27 Magnetic Field and Magnetic Forces

Chapter 27 Magnetic Field and Magnetic Forces Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged

More information

VIII. Magnetic Fields - Worked Examples

VIII. Magnetic Fields - Worked Examples MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.0 Spring 003 VIII. Magnetic Fields - Worked Examples Example : Rolling rod A rod with a mass m and a radius R is mounted on two parallel rails

More information

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D

1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be

More information

The Solar Wind. Earth s Magnetic Field p.1/15

The Solar Wind. Earth s Magnetic Field p.1/15 The Solar Wind 1. The solar wind is a stream of charged particles - a plasma - from the upper atmosphere of the sun consisting of electrons and protons with energies of 1 kev. 2. The particles escape the

More information

Nowadays we know that magnetic fields are set up by charges in motion, as in

Nowadays we know that magnetic fields are set up by charges in motion, as in 6 Magnetostatics 6.1 The magnetic field Although the phenomenon of magnetism was known about as early as the 13 th century BC, and used in compasses it was only in 1819 than Hans Oersted recognised that

More information

Review Questions PHYS 2426 Exam 2

Review Questions PHYS 2426 Exam 2 Review Questions PHYS 2426 Exam 2 1. If 4.7 x 10 16 electrons pass a particular point in a wire every second, what is the current in the wire? A) 4.7 ma B) 7.5 A C) 2.9 A D) 7.5 ma E) 0.29 A Ans: D 2.

More information

Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering. Part A

Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering. Part A Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering Part A 1. Four particles follow the paths shown in Fig. 32-33 below as they pass through the magnetic field there. What can one conclude

More information

Physics Notes for Class 12 Chapter 4 Moving Charges and Magnetrism

Physics Notes for Class 12 Chapter 4 Moving Charges and Magnetrism 1 P a g e Physics Notes for Class 12 Chapter 4 Moving Charges and Magnetrism Oersted s Experiment A magnetic field is produced in the surrounding of any current carrying conductor. The direction of this

More information

University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P. Pebler

University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P. Pebler University of California, Berkeley Physics H7B Spring 1999 (Strovink) SOLUTION TO PROBLEM SET 10 Solutions by P Pebler 1 Purcell 66 A round wire of radius r o carries a current I distributed uniformly

More information

Lecture L5 - Other Coordinate Systems

Lecture L5 - Other Coordinate Systems S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

More information

Chapter 26 Magnetism

Chapter 26 Magnetism What is the fundamental hypothesis of science, the fundamental philosophy? [It is the following:] the sole test of the validity of any idea is experiment. Richard P. Feynman 26.1 The Force on a Charge

More information

AP2 Magnetism. (c) Explain why the magnetic field does no work on the particle as it moves in its circular path.

AP2 Magnetism. (c) Explain why the magnetic field does no work on the particle as it moves in its circular path. A charged particle is projected from point P with velocity v at a right angle to a uniform magnetic field directed out of the plane of the page as shown. The particle moves along a circle of radius R.

More information

Chapter 22 Magnetism

Chapter 22 Magnetism 22.6 Electric Current, Magnetic Fields, and Ampere s Law Chapter 22 Magnetism 22.1 The Magnetic Field 22.2 The Magnetic Force on Moving Charges 22.3 The Motion of Charged particles in a Magnetic Field

More information

Magnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise.

Magnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise. Magnetism 1. An electron which moves with a speed of 3.0 10 4 m/s parallel to a uniform magnetic field of 0.40 T experiences a force of what magnitude? (e = 1.6 10 19 C) a. 4.8 10 14 N c. 2.2 10 24 N b.

More information

Objectives for the standardized exam

Objectives for the standardized exam III. ELECTRICITY AND MAGNETISM A. Electrostatics 1. Charge and Coulomb s Law a) Students should understand the concept of electric charge, so they can: (1) Describe the types of charge and the attraction

More information

Physics 2212 GH Quiz #4 Solutions Spring 2015

Physics 2212 GH Quiz #4 Solutions Spring 2015 Physics 1 GH Quiz #4 Solutions Spring 15 Fundamental Charge e = 1.6 1 19 C Mass of an Electron m e = 9.19 1 31 kg Coulomb constant K = 8.988 1 9 N m /C Vacuum Permittivity ϵ = 8.854 1 1 C /N m Earth s

More information

11. Sources of Magnetic Fields

11. Sources of Magnetic Fields 11. Sources of Magnetic Fields S. G. Rajeev February 24, 2009 1 Magnetic Field Due to a Straight Wire We saw that electric currents produce magnetic fields. The simplest situation is an infinitely long,

More information

Force on Moving Charges in a Magnetic Field

Force on Moving Charges in a Magnetic Field [ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

More information

Magnetic Field and Magnetic Forces

Magnetic Field and Magnetic Forces Chapter 27 Magnetic Field and Magnetic Forces PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 27 Magnets

More information

Physics 2220 Module 09 Homework

Physics 2220 Module 09 Homework Physics 2220 Module 09 Homework 01. A potential difference of 0.050 V is developed across the 10-cm-long wire of the figure as it moves though a magnetic field perpendicular to the page. What are the strength

More information

Lesson 12: Magnetic Forces and Circular Motion!

Lesson 12: Magnetic Forces and Circular Motion! Lesson 12: Magnetic Forces and Circular Motion If a magnet is placed in a magnetic field, it will experience a force. Types of magnets: Direction of the force on a permanent magnet: Direction of the force

More information

Fall 12 PHY 122 Homework Solutions #8

Fall 12 PHY 122 Homework Solutions #8 Fall 12 PHY 122 Homework Solutions #8 Chapter 27 Problem 22 An electron moves with velocity v= (7.0i - 6.0j)10 4 m/s in a magnetic field B= (-0.80i + 0.60j)T. Determine the magnitude and direction of the

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 29a. Electromagnetic Induction Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the

More information

Vector surface area Differentials in an OCS

Vector surface area Differentials in an OCS Calculus and Coordinate systems EE 311 - Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals

More information

My lecture slides are posted at Information for Physics 112 midterm, Wednesday, May 2

My lecture slides are posted at  Information for Physics 112 midterm, Wednesday, May 2 My lecture slides are posted at http://www.physics.ohio-state.edu/~humanic/ Information for Physics 112 midterm, Wednesday, May 2 1) Format: 10 multiple choice questions (each worth 5 points) and two show-work

More information

2 A Dielectric Sphere in a Uniform Electric Field

2 A Dielectric Sphere in a Uniform Electric Field Dielectric Problems and Electric Susceptability Lecture 10 1 A Dielectric Filled Parallel Plate Capacitor Suppose an infinite, parallel plate capacitor with a dielectric of dielectric constant ǫ is inserted

More information

THE MAGNETIC FIELD. 9. Magnetism 1

THE MAGNETIC FIELD. 9. Magnetism 1 THE MAGNETIC FIELD Magnets always have two poles: north and south Opposite poles attract, like poles repel If a bar magnet is suspended from a string so that it is free to rotate in the horizontal plane,

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

Physics 9 Fall 2009 Homework 2 - Solutions

Physics 9 Fall 2009 Homework 2 - Solutions Physics 9 Fall 009 Homework - s 1. Chapter 7 - Exercise 5. An electric dipole is formed from ±1.0 nc charges spread.0 mm apart. The dipole is at the origin, oriented along the y axis. What is the electric

More information

Force on a square loop of current in a uniform B-field.

Force on a square loop of current in a uniform B-field. Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis

More information

PHYS 155: Final Tutorial

PHYS 155: Final Tutorial Final Tutorial Saskatoon Engineering Students Society eric.peach@usask.ca April 13, 2015 Overview 1 2 3 4 5 6 7 Tutorial Slides These slides have been posted: sess.usask.ca homepage.usask.ca/esp991/ Section

More information

Chapter 33. The Magnetic Field

Chapter 33. The Magnetic Field Chapter 33. The Magnetic Field Digital information is stored on a hard disk as microscopic patches of magnetism. Just what is magnetism? How are magnetic fields created? What are their properties? These

More information

Chapter 22: Electric Flux and Gauss s Law

Chapter 22: Electric Flux and Gauss s Law 22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

More information

E/M Experiment: Electrons in a Magnetic Field.

E/M Experiment: Electrons in a Magnetic Field. E/M Experiment: Electrons in a Magnetic Field. PRE-LAB You will be doing this experiment before we cover the relevant material in class. But there are only two fundamental concepts that you need to understand.

More information

Magnetic Fields and Forces. AP Physics B

Magnetic Fields and Forces. AP Physics B Magnetic ields and orces AP Physics acts about Magnetism Magnets have 2 poles (north and south) Like poles repel Unlike poles attract Magnets create a MAGNETIC IELD around them Magnetic ield A bar magnet

More information

Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j

Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD. q j Seminar 4: CHARGED PARTICLE IN ELECTROMAGNETIC FIELD Introduction Let take Lagrange s equations in the form that follows from D Alembert s principle, ) d T T = Q j, 1) dt q j q j suppose that the generalized

More information

Profs. A. Petkova, A. Rinzler, S. Hershfield. Exam 2 Solution

Profs. A. Petkova, A. Rinzler, S. Hershfield. Exam 2 Solution PHY2049 Fall 2009 Profs. A. Petkova, A. Rinzler, S. Hershfield Exam 2 Solution 1. Three capacitor networks labeled A, B & C are shown in the figure with the individual capacitor values labeled (all units

More information

Magnetic Fields; Sources of Magnetic Field

Magnetic Fields; Sources of Magnetic Field This test covers magnetic fields, magnetic forces on charged particles and current-carrying wires, the Hall effect, the Biot-Savart Law, Ampère s Law, and the magnetic fields of current-carrying loops

More information

CHARGE TO MASS RATIO OF THE ELECTRON

CHARGE TO MASS RATIO OF THE ELECTRON CHARGE TO MASS RATIO OF THE ELECTRON In solving many physics problems, it is necessary to use the value of one or more physical constants. Examples are the velocity of light, c, and mass of the electron,

More information

Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 12 Solutions

Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 12 Solutions Concept Check (top) Pearson Physics Level 30 Unit VI Forces and Fields: Chapter 1 Solutions Student Book page 583 Concept Check (bottom) The north-seeking needle of a compass is attracted to what is called

More information

Chapter 6 Circular Motion

Chapter 6 Circular Motion Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

More information

Lab 4: Magnetic Force on Electrons

Lab 4: Magnetic Force on Electrons Lab 4: Magnetic Force on Electrons Introduction: Forces on particles are not limited to gravity and electricity. Magnetic forces also exist. This magnetic force is known as the Lorentz force and it is

More information

CET Moving Charges & Magnetism

CET Moving Charges & Magnetism CET 2014 Moving Charges & Magnetism 1. When a charged particle moves perpendicular to the direction of uniform magnetic field its a) energy changes. b) momentum changes. c) both energy and momentum

More information

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.

* Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No. * Biot Savart s Law- Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying

More information

Let s first see how precession works in quantitative detail. The system is illustrated below: ...

Let s first see how precession works in quantitative detail. The system is illustrated below: ... lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

More information

Home Work 9. i 2 a 2. a 2 4 a 2 2

Home Work 9. i 2 a 2. a 2 4 a 2 2 Home Work 9 9-1 A square loop of wire of edge length a carries current i. Show that, at the center of the loop, the of the magnetic field produced by the current is 0i B a The center of a square is a distance

More information

Moving Charge and Faraday s Law - Lecture 2

Moving Charge and Faraday s Law - Lecture 2 Moving Charge and Faraday s Law - Lecture 2 1 Introduction The static equation, E = 0 holds only for charge at rest. However, it will be found that E = 0 does not transform properly for charge in motion,

More information

Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements)

Test - A2 Physics. Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Test - A2 Physics Primary focus Magnetic Fields - Secondary focus electric fields (including circular motion and SHM elements) Time allocation 40 minutes These questions were ALL taken from the June 2010

More information

Transformed E&M I homework. Divergence and Curl of B (Ampereʼs Law) (Griffiths Chapter 5)

Transformed E&M I homework. Divergence and Curl of B (Ampereʼs Law) (Griffiths Chapter 5) Transformed E&M I homework Divergence and Curl of B (Ampereʼs Law) (Griffiths Chapter 5) Divergence and curl of B (Connections between E and B, Ampere s Law) Question 1. B of cylinder with hole Pollack

More information

Eðlisfræði 2, vor 2007

Eðlisfræði 2, vor 2007 [ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

More information

1 of 7 4/13/2010 8:05 PM

1 of 7 4/13/2010 8:05 PM Chapter 33 Homework Due: 8:00am on Wednesday, April 7, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy [Return to Standard Assignment View] Canceling a Magnetic Field

More information

Chapter 19: Magnetic Forces and Fields

Chapter 19: Magnetic Forces and Fields Chapter 19: Magnetic Forces and Fields Magnetic Fields Magnetic Force on a Point Charge Motion of a Charged Particle in a Magnetic Field Crossed E and B fields Magnetic Forces on Current Carrying Wires

More information

Chapter 20. Magnetic Forces and Magnetic Fields

Chapter 20. Magnetic Forces and Magnetic Fields Chapter 20 Magnetic Forces and Magnetic Fields Magnetic Fields The most familiar example of magnetism for most people is a magnet. Every magnet has two poles, North and South --> called this since if the

More information

Candidate Number. General Certificate of Education Advanced Level Examination June 2010

Candidate Number. General Certificate of Education Advanced Level Examination June 2010 entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 1 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Friday 18

More information

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

More information

4.1.Motion Of Charged Particles In Electric And Magnetic Field Motion Of Charged Particles In An Electric Field

4.1.Motion Of Charged Particles In Electric And Magnetic Field Motion Of Charged Particles In An Electric Field 4.1.Motion Of Charged Particles In Electric And Magnetic Field 4.1.1. Motion Of Charged Particles In An Electric Field A charged particle in an electric field will experience an electric force due to the

More information

Modern Physics Laboratory e/m with Teltron Deflection Tube

Modern Physics Laboratory e/m with Teltron Deflection Tube Modern Physics Laboratory e/m with Teltron Deflection Tube Josh Diamond & John Cummings Fall 2010 Abstract The deflection of an electron beam by electric and magnetic fields is observed, and the charge

More information

Physics 12 Study Guide: Electromagnetism Magnetic Forces & Induction. Text References. 5 th Ed. Giancolli Pg

Physics 12 Study Guide: Electromagnetism Magnetic Forces & Induction. Text References. 5 th Ed. Giancolli Pg Objectives: Text References 5 th Ed. Giancolli Pg. 588-96 ELECTROMAGNETISM MAGNETIC FORCE AND FIELDS state the rules of magnetic interaction determine the direction of magnetic field lines use the right

More information

Electromagnetic Induction. Physics 231 Lecture 9-1

Electromagnetic Induction. Physics 231 Lecture 9-1 Electromagnetic Induction Physics 231 Lecture 9-1 Induced Current Past experiments with magnetism have shown the following When a magnet is moved towards or away from a circuit, there is an induced current

More information

Conceptual: 1, 3, 5, 6, 8, 16, 18, 19. Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65. Conceptual Questions

Conceptual: 1, 3, 5, 6, 8, 16, 18, 19. Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65. Conceptual Questions Conceptual: 1, 3, 5, 6, 8, 16, 18, 19 Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65 Conceptual Questions 1. The magnetic field cannot be described as the magnetic force per unit charge

More information

APPLIED MATHEMATICS ADVANCED LEVEL

APPLIED MATHEMATICS ADVANCED LEVEL APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications

More information

Physics of fusion power. Lecture 6: Conserved quantities / Mirror device / tokamak

Physics of fusion power. Lecture 6: Conserved quantities / Mirror device / tokamak Physics of fusion power Lecture 6: Conserved quantities / Mirror device / tokamak Reminder Perpendicular forces lead to drifts of the particles Electric field acceleration Inertia connected with a change

More information

Chapter 19 Magnetism Magnets Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Like poles

Chapter 19 Magnetism Magnets Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Like poles Chapter 19 Magnetism Magnets Poles of a magnet are the ends where objects are most strongly attracted Two poles, called north and south Like poles repel each other and unlike poles attract each other Similar

More information

Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

More information

PSS 27.2 The Electric Field of a Continuous Distribution of Charge

PSS 27.2 The Electric Field of a Continuous Distribution of Charge Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

More information

My Website:

My Website: PH203 Recitation Week 09 Problem Set Spring 2015 Ryan Scheirer Email: scheirer@onid.orst.edu My Website: http://people.oregonstate.edu/~scheirer/ph203_rec.html Problem 01 For the following questions, use

More information

Candidate Number. General Certificate of Education Advanced Level Examination June 2012

Candidate Number. General Certificate of Education Advanced Level Examination June 2012 entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 212 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Monday

More information

Magnetism Conceptual Questions. Name: Class: Date:

Magnetism Conceptual Questions. Name: Class: Date: Name: Class: Date: Magnetism 22.1 Conceptual Questions 1) A proton, moving north, enters a magnetic field. Because of this field, the proton curves downward. We may conclude that the magnetic field must

More information

Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y

Projectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y Projectile Motion! An object may move in both the x and y directions simultaneously! The form of two-dimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion! The

More information

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014

arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic

More information

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 30 - Magnetic Fields and Torque. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 30 - Magnetic Fields and Torque A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Objectives: After completing this module, you should

More information

Lecture PowerPoints. Chapter 20 Physics: Principles with Applications, 7 th edition Giancoli

Lecture PowerPoints. Chapter 20 Physics: Principles with Applications, 7 th edition Giancoli Lecture PowerPoints Chapter 20 Physics: Principles with Applications, 7 th edition Giancoli This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching

More information

( )( 10!12 ( 0.01) 2 2 = 624 ( ) Exam 1 Solutions. Phy 2049 Fall 2011

( )( 10!12 ( 0.01) 2 2 = 624 ( ) Exam 1 Solutions. Phy 2049 Fall 2011 Phy 49 Fall 11 Solutions 1. Three charges form an equilateral triangle of side length d = 1 cm. The top charge is q = - 4 μc, while the bottom two are q1 = q = +1 μc. What is the magnitude of the net force

More information

1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius?

1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius? CHAPTER 3 SOURCES O THE MAGNETC ELD 1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 8 µ T at the loop center. What is the loop radius? Equation 3-3, with

More information

Magnetic Forces and Magnetic Fields

Magnetic Forces and Magnetic Fields 1 Magnets Magnets are metallic objects, mostly made out of iron, which attract other iron containing objects (nails) etc. Magnets orient themselves in roughly a north - south direction if they are allowed

More information

Magnetic Fields. David J. Starling Penn State Hazleton PHYS 212

Magnetic Fields. David J. Starling Penn State Hazleton PHYS 212 Magnetism, as you recall from physics class, is a powerful force that causes certain items to be attracted to refrigerators. - Dave Barry David J. Starling Penn State Hazleton PHYS 212 Objectives (a) Determine

More information

Physics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5

Physics 112 Homework 5 (solutions) (2004 Fall) Solutions to Homework Questions 5 Solutions to Homework Questions 5 Chapt19, Problem-2: (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields in Figure P19.2, as shown. (b) Repeat

More information

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.

The purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law. 260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this

More information

Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P

Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P Q28.1 A positive point charge is moving to the right. The magnetic field that the point charge produces at point P (see diagram below) P r + v r A. points in the same direction as v. B. points from point

More information

Worked solutions Chapter 9 Magnets and electricity

Worked solutions Chapter 9 Magnets and electricity 9.1 Fundamentals of magnetism 1 A magnetic field exists at any point in space where a magnet or magnetic material (e.g. iron, nickel, cobalt) will experience a magnetic force. 2 C. The magnetic force between

More information

mv = ev ebr Application: circular motion of moving ions In a uniform magnetic field: The mass spectrometer KE=PE magnitude of electron charge

mv = ev ebr Application: circular motion of moving ions In a uniform magnetic field: The mass spectrometer KE=PE magnitude of electron charge 1.4 The Mass Spectrometer Application: circular motion of moving ions In a uniform magnetic field: The mass spectrometer mv r qb mv eb magnitude of electron charge 1 mv ev KEPE v 1 mv ebr m v e r m B m

More information

PHYS 110B - HW #2 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased

PHYS 110B - HW #2 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased PHYS 11B - HW # Fall 5, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased [1.] Problem 7. from Griffiths A capacitor capacitance, C) is charged

More information

Date: Deflection of an Electron in a Magnetic Field

Date: Deflection of an Electron in a Magnetic Field Name: Partners: Date: Deflection of an Electron in a Magnetic Field Purpose In this lab, we use a Cathode Ray Tube (CRT) to measure the effects of an electric and magnetic field on the motion of a charged

More information

Then the second equation becomes ³ j

Then the second equation becomes ³ j Magnetic vector potential When we derived the scalar electric potential we started with the relation r E = 0 to conclude that E could be written as the gradient of a scalar potential. That won t work for

More information

Electromagnetism - Lecture 2. Electric Fields

Electromagnetism - Lecture 2. Electric Fields Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric

More information

Magnetostatics II. Lecture 24: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay

Magnetostatics II. Lecture 24: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetostatics II Lecture 4: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Magnetic field due to a solenoid and a toroid A solenoid is essentially a long current loop

More information

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

Chapter 29. Magnetic Fields

Chapter 29. Magnetic Fields Chapter 29 Magnetic Fields A Partial History of Magnetism 13 th century BC Chinese used a compass 800 BC Uses a magnetic needle Probably an invention of Arabic or Indian origin Greeks Discovered magnetite

More information

Problem Set V Solutions

Problem Set V Solutions Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3

More information

Physics 9 Fall 2009 Homework 7 - Solutions

Physics 9 Fall 2009 Homework 7 - Solutions Physics 9 Fall 009 Homework 7 - s 1. Chapter 33 - Exercise 10. At what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

Chapter 5. Magnetostatics and Electromagnetic Induction

Chapter 5. Magnetostatics and Electromagnetic Induction Chapter 5. Magnetostatics and Electromagnetic Induction 5.1 Magnetic Field of Steady Currents The Lorentz force law The magnetic force in a charge q, moving with velocity v in a magnetic field B in a magnetic

More information

" - angle between l and a R

 - angle between l and a R Magnetostatic Fields According to Coulomb s law, any distribution of stationary charge produces a static electric field (electrostatic field). The analogous equation to Coulomb s law for electric fields

More information

Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

More information

Electric Forces & Fields, Gauss s Law, Potential

Electric Forces & Fields, Gauss s Law, Potential This test covers Coulomb s Law, electric fields, Gauss s Law, electric potential energy, and electric potential, with some problems requiring a knowledge of basic calculus. Part I. Multiple Choice +q +2q

More information

) 0.7 =1.58 10 2 N m.

) 0.7 =1.58 10 2 N m. Exam 2 Solutions Prof. Paul Avery Prof. Andrey Korytov Oct. 29, 2014 1. A loop of wire carrying a current of 2.0 A is in the shape of a right triangle with two equal sides, each with length L = 15 cm as

More information

MOVING CHARGES AND MAGNETISM

MOVING CHARGES AND MAGNETISM MOVING CHARGES AND MAGNETISM 1. A circular Coil of 50 turns and radius 0.2m carries of current of 12A Find (a). magnetic field at the centre of the coil and (b) magnetic moment associated with it. 3 scores

More information

Concept Review. Physics 1

Concept Review. Physics 1 Concept Review Physics 1 Speed and Velocity Speed is a measure of how much distance is covered divided by the time it takes. Sometimes it is referred to as the rate of motion. Common units for speed or

More information