Electromagnetic Waves

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Electromagnetic Waves"

Transcription

1 May 4, J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 7

2 Maxwell Equations A basic feature of Maxwell equations for the EM field is the existence of travelling wave solutions which represent the transport of energy from one point to another. The simplest and most fundamental EM waves are transverse, plane waves. In a region of space where there are no free sources (ρ = 0, J = 0), Maxwell s equations reduce to a simple form given E = 0, E 1 B + c t = 0 B = 0, B µɛ c where D and H are given by relations E t = 0 (1) D = ɛ E and H = 1 µ B (2) where ɛ is the electric permittivity and µ the magnetic permeability which assumed to be independent of the frequency.

3 Plane Maxwell s equations can be written as 2 B µɛ c 2 2 B t 2 = 0 and 2 E µɛ c 2 2 E t 2 = 0 (3) In other words each component of B and E obeys a wave equation of the form: 2 u 1 2 u v 2 t 2 = 0 where v = c (4) µɛ is a constant with dimensions of velocity characteristic of the medium. The wave equation admits admits plane-wave solutions: u = e i k x iωt E( x, t) = Ee ik n x iωt and B( x, t) = Be ik n x iωt (5) (6) where the relation between the frequency ω and the wave vector k is k = ω v = µɛ ω c or k k = ( ω v ) 2 (7) also the vectors n, E and B are constant in time and space.

4 If we consider waves propagating in one direction, say x-direction then the fundamental solution is: u(x, t) = Ae ik(x vt) + Be ik(x+vt) (8) which represents waves traveling to the right and to the left with propagation velocities v which is called phase velocity of the wave. From the divergence relations of (1) by applying (6) we get n E = 0 and n B = 0 (9) This means that E (or E) and B (or B) are both perpendicular to the direction of propagation n. Such a wave is called transverse wave. The curl equations provide a further restriction B = µɛ n E and E = 1 µɛ n B (10) The combination of equations (9) and (10) suggests that the vectors n, E and B form an orthonormal set. Also, if n is real, then (10) implies that that E and B have the same phase.

5 It is then useful to introduce a set of real mutually orthogonal unit vectors ( ɛ 1, ɛ 2, n). In terms of these unit vectors the field strengths E and B are or E = ɛ 1 E 0, B = ɛ2 µɛe0 (11) E = ɛ 2 E 0, B = ɛ1 µɛe 0 (12) E 0 and E 0 are constants, possibly complex. In other words the most general way to write the electric/magnetic field vector is: E = (E 0 ɛ 1 + E 0 ɛ 2 )e ik n x iωt (13) B = µɛ(e 0 ɛ 2 E 0 ɛ 1 )e ik n x iωt (14)

6 Thus the wave described by (6) and (11) or (12) is a transverse wave propagating in the direction n. Or that E and B are oscillating in a plane perpendicular to the wave vector k, determining the direction of propagation of the wave. The energy flux of EM waves is described by the real part of the complex Poynting vector 1 c S = E 2 4π H = 1 c [ ER H 2 4π R + E I H I + i( EI H R E R H )] I where E and H are the measured fields at the point where S is evaluated. 2 2 Note : we use the magnetic induction H because although B is the applied induction, the actual field that carries the energy and momentum in media is H.

7 The time averaged flux of energy is: c ɛ S = 8π µ E 0 2 n (15) The total time averaged density (and not just the energy density associated with the electric field component) is: u = 1 (ɛ E 16π E + 1µ ) B B = ɛ 8π E 0 2 (16) The ratio of the magnitude of (15) to (16) is the speed of energy flow i.e. v = c/ µɛ. 3 (Prove the above relations) Project: What will happen if n is not real? What type of waves you will get? What will be the form of E? 3 Note: To prove the above relations use cos 2 x = 1/2 and since E R = ( E + E )/2 we get E 2 R = E E /2.

8 Linear and Circular Polarization of EM Waves The plane wave (6) and (11) is a wave with its electric field vector always in the direction ɛ 1. Such a wave is said to be linearly polarized with polarization vector ɛ 1. The wave described by (12) is linearly polarized with polarization vector ɛ 2 and is linearly independent of the first. The two waves : E 1 = ɛ 1 E 1 e i k x iωt, E2 = ɛ 2 E 2 e i k x iωt with (17) B i = µɛ k E i, i = 1, 2 k Can be combined to give the most general homogeneous plane waves propagating in the direction k = k n, E( x, t) = ( ɛ 1 E 1 + ɛ 2 E 2 ) e i k x iωt (18) [ E( x, t) = ɛ 1 E 1 + ɛ 2 E 2 e i(φ2 φ1)] e i k x iωt+iφ 1 (19) The amplitudes E 1 = E 1 e iφ1 and E 2 = E 2 e iφ2 are complex numbers in order to allow the possibility of a phase difference between waves of different polarization.

9 LINEARLY POLARIZED If the amplitudes E 1 = E 1 e iφ1 and E 2 = E 2 e iφ2 have the same phase (18) represents a linearly polarized wave with the polarization vector making an angle θ = tan 1 (R(E 2 )/I(E 1 )) (which remains constant as the field evolves in space and time) with ɛ 1 and magnitude E = E E 2 2. ELLIPTICALLY POLARIZED If E 1 and E 2 have the different phase the wave (18) is elliptically polarized and the electric vector rotates around k.

10 Circular Polarization E 1 = E 2 = E 0 φ 1 φ 2 = ±π/2 and the wave becomes E( x, t) = E 0 ( ɛ 1 ± i ɛ 2 ) e i k x iωt (20) At a fixed point in space, the fields are such that the electric vector is constant in magnitude, but sweeps around in a circle at a frequency ω. The components of the electric field, obtained by taking the real part of (20) E x ( x, t) = E 0 cos(kz ωt), E y ( x, t) = E 0 cos(kz ωt) (21) For the upper sign ( ɛ 1 + i ɛ 2 ) the rotation is counter-clockwise when the observer is facing into the oncoming wave. The wave is called left circularly polarized in optics while in modern physics such a wave is said to have positive helicity. For the lower sign ( ɛ 1 i ɛ 2 ) the wave is right circularly polarized or it has negative helicity.

11 Elliptically Polarized EM Waves An alternative general expression for E can be given in terms of the complex orthogonal vectors with properties ɛ ± = 1 2 ( ɛ 1 ± i ɛ 2 ) (22) ɛ ± ɛ = 0, ɛ ± ɛ 3 = 0, ɛ ± ɛ ± = 1. (23) Then the general representation of the electric vector E( x, t) = (E + ɛ + + E ɛ ) e i k x iωt (24) where E and E + are complex amplitudes If E and E + have different amplitudes but the same phase eqn (24) represents an elliptically polarized wave with principle axes of the ellipse in the directions of ɛ 1 and ɛ 2. The ratio of the semimajor to semiminor axis is (1 + r)/(1 r), where E /E + = r.

12 The ratio of the semimajor to semiminor axis is (1 + r)/(1 r), where E /E + = r. If the amplitudes have a phase difference between them E /E + = re iα, then the ellipse traced out by the E vector has its axes rotated by an angle α/2. Figure: The figure shows the general case of elliptical polarization and the ellipses traced out by both E and B at a given point in space. Note : For r = ±1 we get back to a linearly polarized wave.

13 Polarization Figure: The figure shows the linear, circular and elliptical polarization

14 Stokes Parameters The polarization content of an EM wave is known if it can be written in the form of either (18) or (24) with known coefficients (E 1, E 2 ) or (E, E + ). In practice, the converse problem arises i.e. given a wave of the form (6), how can we determine from observations on the beam the state of polarization? A useful tool for this are the four Stokes parameters. These are quadratic in the field strength and can be determined through intensity measurements only. Their measurements determines completely the state of polarization of the wave. For a wave propagating in the z-direction the scalar products ɛ 1 E, ɛ 2 E, ɛ + E, ɛ E (25) are the amplitudes of radiation respectively, with linear polarization in the x-direction, linear polarization in the y-direction, positive helicity and negative helicity. The squares of these amplitudes give a measure of the intensity of each type of polarization. The phase information can be taken by using cross products

15 In terms of the linear polarization bases ( ɛ 1, ɛ 2 ), the Stokes parameters are: s 0 = ɛ 1 E 2 + ɛ 2 E 2 = a a 2 2 s 1 = ɛ 1 E 2 ɛ 2 E 2 = a1 2 a2 2 [ s 2 = 2R ( ɛ 1 E) ( ɛ 1 E) ] = 2a 1 a 2 cos(δ 1 δ 2 ) (26) [ s 3 = 2I ( ɛ 1 E) ( ɛ 1 E) ] = 2a 1 a 2 sin(δ 1 δ 2 ) where we defined the coefficients of (18) or (24) as magnitude times a phase factor: E 1 = a 1 e iδ1, E 2 = a 2 e iδ2, E + = a + e iδ+, E = a e iδ (27) Here s 0 and s 1 contain information regarding the amplitudes of linear polarization, whereas s 2 and s 3 say something about the phases. Knowing these parameters (e.g by passing a wave through perpendicular polarization filters) is sufficient for us to determine the amplitudes and relative phases of the field components.

16 Stokes Parameters In terms of the linear polarization bases ( ɛ +, ɛ ), the Stokes parameters are: s 0 = ɛ + E 2 + ɛ E 2 = a+ 2 + a 2 [ s 1 = 2R ( ɛ + E) ( ɛ E) ] = 2a + a cos(δ δ + ) (28) [ s 2 = 2I ( ɛ + E) ( ɛ E) ] = 2a + a sin(δ δ + ) s 3 = ɛ + E 2 ɛ E 2 = a 2 + a 2 Notice an interesting rearrangement of roles of the Stokes parameters with respect to the two bases. The four Stokes parameters are not independent since they depend on only 3 quantities a 1, a 2 and δ 1 δ 2. They satisfy the relation s 2 0 = s s s 2 3. (29)

17 Reflection & Refraction of EM Waves The reflection and refraction of light at a plane surface between two media of different dielectric properties are familiar phenomena. The various aspects of the phenomena divide themselves into two classes Kinematic properties: Angle of reflection = angle of incidence sin i Snell s law: sin r = n n where i, r are the angles of incidence and refraction, while n, n are the corresponding indices of refraction. Dynamic properties: Intensities of reflected and refracted radiation Phase changes and polarization The kinematic properties follow from the wave nature of the phenomena and the need to satisfy certain boundary conditions (BC). But not on the detailed nature of the waves or the boundary conditions. The dynamic properties depend entirely on the specific nature of the EM fields and their boundary conditions.

18 Figure: Incident wave k strikes plane interface between different media, giving rise to a reflected wave k and a refracted wave k. The media below and above the plane z = 0 have permeabilities and dielectric constants µ, ɛ and µ, ɛ respectively. The indices of refraction are n = µɛ and n = µ ɛ.

19 According to eqn (18) the 3 waves are: INCIDENT REFRACTED E = E 0 e i k x iωt, B = µɛ k E k (30) REFLECTED E = E 0e i k x iωt, B = µ ɛ k E k (31) E = E 0 e i k x iωt, B = µ ɛ k E k (32) The wave numbers have magnitudes: k = k = k = ω µɛ, c k = k = ω c µ ɛ (33)

20 AT the boundary z = 0 the BC must be satisfied at all points on the plane at all times, i.e. the spatial & time variation of all fields must be the same at z = 0. Thus the phase factors must be equal at z = 0 ( ) ( ) ( ) k x = k x = k x (34) z=0 independent of the nature of the boundary conditions. Eqn (34) contains the kinematic aspects of reflection and refraction. Note that all 3 wave vectors must lie in a plane. From the previous figure we get k sin i = k sin r = k sin r (35) Since k = k, we find that i = r ; the angle of incidence equals the angle of reflection. Snell s law is: z=0 sin i sin r = k k = µ ɛ µɛ = n n z=0 (36)

21 Reflection & Refraction of EM Waves The dynamic properties are contained in the boundary conditions : normal components of D = ɛe and B are continuous tangential components of E and H = [c/(ωµ)] k E are continuous In terms of fields (30)-(32) these boundary conditions at z = 0 are: [ 1 µ [ ( ɛ E0 + E ) ] 0 ɛ E 0 n = 0 [ k E0 + k E 0 k E ] 0 n = 0 ( E0 + E 0 E ) 0 n = 0 (37) ( k E0 + k E ) 0 1 ( k µ E 0) ] n = 0 Two separate situations, the incident plane wave is linearly polarized : The polarization vector is perpendicular to the plane of incidence (the plane defined by k and n ). The polarization vector is parallel to the plane of incidence. The case of arbitrary elliptic polarization can be obtained by appropriate linear combinations of the two results.

22 E : Perpendicular to the plane of incidence Since the E-fields are parallel to the surface the 1st BC of (38) yields nothing The 3rd and 4th of of (38) give (how?): E 0 + E 0 E 0 = 0 ɛ µ (E 0 E 0 ɛ ) cos i µ E 0 cos r = 0 (38) The 2nd, using Snell s law, duplicates the 3rd. (prove all the above statements) Figure: Reflection and refraction with polarization perpendicular to the plane of incidence. All the E-fields shown directed away from the viewer.

23 E : Perpendicular to the plane of incidence The relative amplitudes of the refracted and reflected waves can be found from (38) E 0 [ ] 2n cos i 2 2 sin r cos i = = E 0 n cos i + µ µ n 2 n 2 sin 2 i 1 + µ tan i = sin(i + r) µ tan r µ=µ E 0 = n cos i µ µ n 2 n 2 sin 2 i E 0 n cos i + µ µ n 2 n 2 sin 2 i = 1 µ tan i µ tan r 1 + µ tan i µ tan r = [ ] sin(r i) (39) sin(i + r) µ=µ Note that n 2 n 2 sin 2 i = n cos r but Snell s law has been used to express it in terms of the angle of incidence. For optical frequencies it is usually permitted to put µ = µ.

24 E : Parallel to the plane of incidence Boundary conditions involved: normal E : 1st eqn in (38) tangential E : 3rd eqn in (38) tangential B : 4th eqn in (38) The last two demand that (E 0 E 0 ) cos i E 0 cos r = 0 ɛ µ (E 0 + E 0 ɛ ) µ E 0 = 0 (40) Figure: Reflection and refraction with polarization parallel to the plane of incidence.

25 E : Parallel to the plane of incidence The condition that normal E is continuous, plus Snell s law, merely dublicates the 2nd of the previous equations. The relative amplitudes of refracted and reflected fields are therefore (how?) E 0 E 0 = E 0 E 0 = 2nn cos i µ µ n 2 cos i + n n 2 n 2 sin 2 i 2 n ɛ [ ] nɛ 2 sin r cos i = 1 + ɛ tan i = sin(i + r) cos(i r) ɛ tan r µ=µ µ µ n 2 cos i n n 2 n 2 sin 2 i µ µ n 2 cos i + n n 2 n 2 sin 2 i = 1 ɛ tan i [ ] ɛ tan r tan(i r) 1 + ɛ tan i = (41) tan(i + r) ɛ tan r µ=µ

26 Normal incidence i = 0 For normal incidence i = 0 both (39) and (41) reduce to (how?) E 0 E 0 = µɛ µ ɛ 2n n + n E 0 E 0 = µɛ µ ɛ µɛ µ ɛ n n n + n (42) EXERCISES: What are the conditions for: Total reflection Total transmision

Electromagnetism - Lecture 8. Maxwell s Equations

Electromagnetism - Lecture 8. Maxwell s Equations Electromagnetism - Lecture 8 Maxwell s Equations Continuity Equation Displacement Current Modification to Ampère s Law Maxwell s Equations in Vacuo Solution of Maxwell s Equations Introduction to Electromagnetic

More information

0.1 Dielectric Slab Waveguide

0.1 Dielectric Slab Waveguide 0.1 Dielectric Slab Waveguide At high frequencies (especially optical frequencies) the loss associated with the induced current in the metal walls is too high. A transmission line filled with dielectric

More information

Fundamentals of Electromagnetic Fields and Waves: I

Fundamentals of Electromagnetic Fields and Waves: I Fundamentals of Electromagnetic Fields and Waves: I Fall 2007, EE 30348, Electrical Engineering, University of Notre Dame Mid Term II: Solutions Please show your steps clearly and sketch figures wherever

More information

MODELING AND MEASURING THE POLARIZATION OF LIGHT:

MODELING AND MEASURING THE POLARIZATION OF LIGHT: MODELING AND MEASURING THE POLARIZATION OF LIGHT: FROM JONES MATRICES TO ELLIPSOMETRY OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results.

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

Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN

Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN Special Relativity and Electromagnetism Yannis PAPAPHILIPPOU CERN United States Particle Accelerator School, University of California - Santa Cruz, Santa Rosa, CA 14 th 18 th January 2008 1 Outline Notions

More information

Light. (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8)

Light. (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8) (Material taken from: Optics, by E. Hecht, 4th Ed., Ch: 1,2, 3, 8) is an Electromagnetic (EM) field arising from the non-uniform motion of charged particles. It is also a form of EM energy that originates

More information

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

PHY411. PROBLEM SET 3

PHY411. PROBLEM SET 3 PHY411. PROBLEM SET 3 1. Conserved Quantities; the Runge-Lenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the

More information

1 Scalars, Vectors and Tensors

1 Scalars, Vectors and Tensors DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

More information

Lecture 13 Electromagnetic Waves Ch. 33

Lecture 13 Electromagnetic Waves Ch. 33 Lecture 13 Electromagnetic Waves Ch. 33 Cartoon Opening Demo Topics Electromagnetic waves Traveling E/M wave - Induced electric and induced magnetic amplitudes Plane waves and spherical waves Energy transport

More information

Section V.4: Cross Product

Section V.4: Cross Product Section V.4: Cross Product Definition The cross product of vectors A and B is written as A B. The result of the cross product A B is a third vector which is perpendicular to both A and B. (Because the

More information

20.1 Revisiting Maxwell s equations

20.1 Revisiting Maxwell s equations Scott Hughes 28 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 20: Wave equation & electromagnetic radiation 20.1 Revisiting Maxwell s equations In our

More information

Figure 1.1 Vector A and Vector F

Figure 1.1 Vector A and Vector F CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

More information

How can I tell what the polarization axis is for a linear polarizer?

How can I tell what the polarization axis is for a linear polarizer? How can I tell what the polarization axis is for a linear polarizer? The axis of a linear polarizer determines the plane of polarization that the polarizer passes. There are two ways of finding the axis

More information

The Stern-Gerlach Experiment

The Stern-Gerlach Experiment Chapter The Stern-Gerlach Experiment Let us now talk about a particular property of an atom, called its magnetic dipole moment. It is simplest to first recall what an electric dipole moment is. Consider

More information

Polarization of Light

Polarization of Light Polarization of Light References Halliday/Resnick/Walker Fundamentals of Physics, Chapter 33, 7 th ed. Wiley 005 PASCO EX997A and EX999 guide sheets (written by Ann Hanks) weight Exercises and weights

More information

Vector Calculus. Chapter GRADIENT

Vector Calculus. Chapter GRADIENT Chapter 6 Vector Calculus References: Skilling, Fundamentals of Electric Waves 2; Lorrain & Corson, Intro. to Electromagnetic Fields and Waves, 1; Hecht, Optics, 3.2, Appendix 1 In 1864, James Clerk Maxwell

More information

2 Session Two - Complex Numbers and Vectors

2 Session Two - Complex Numbers and Vectors PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

More information

ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

More information

Chapter 24 Physical Pendulum

Chapter 24 Physical Pendulum Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

More information

Chapter 13. Gravitation

Chapter 13. Gravitation Chapter 13 Gravitation 13.2 Newton s Law of Gravitation In vector notation: Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G = 6.67

More information

Lecture 1. The nature of electromagnetic radiation.

Lecture 1. The nature of electromagnetic radiation. Lecture 1. The nature of electromagnetic radiation. 1. Basic introduction to the electromagnetic field: Dual nature of electromagnetic radiation Electromagnetic spectrum. Basic radiometric quantities:

More information

Bead moving along a thin, rigid, wire.

Bead moving along a thin, rigid, wire. Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing

More information

Frequency Doubling and Second Order Nonlinear Optics

Frequency Doubling and Second Order Nonlinear Optics Frequency Doubling and Second Order Nonlinear Optics Paul M. Petersen DTU Fotonik, Risø campus Technical University of Denmark, Denmark (email: paul.michael.petersen@risoe.dk) Outline of the talk The first

More information

Electromagnetic waves

Electromagnetic waves Chapter 8 lectromagnetic waves David Morin, morin@physics.harvard.edu The waves we ve dealt with so far in this book have been fairly easy to visualize. Waves involving springs/masses, strings, and air

More information

AP1 Waves. (A) frequency (B) wavelength (C) speed (D) intensity. Answer: (A) and (D) frequency and intensity.

AP1 Waves. (A) frequency (B) wavelength (C) speed (D) intensity. Answer: (A) and (D) frequency and intensity. 1. A fire truck is moving at a fairly high speed, with its siren emitting sound at a specific pitch. As the fire truck recedes from you which of the following characteristics of the sound wave from 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

4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet

4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet 4.4 WAVE CHARACTERISTICS 4.5 WAVE PROPERTIES HW/Study Packet Required: READ Hamper pp 115-134 SL/HL Supplemental: Cutnell and Johnson, pp 473-477, 507-513 Tsokos, pp 216-242 REMEMBER TO. Work through all

More information

The Wave Equation and the Speed of Light

The Wave Equation and the Speed of Light The Wave Equation and the Speed of Light Chapter 1 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 1 Class Outline Maxwell s equations Boundary conditions Poynting s theorem and conservation

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

BCM 6200 - Protein crystallography - I. Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS

BCM 6200 - Protein crystallography - I. Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS BCM 6200 - Protein crystallography - I Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS Elastic X-ray Scattering From classical electrodynamics, the electric field of the electromagnetic

More information

Special Theory of Relativity

Special Theory of Relativity June 1, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 11 Introduction Einstein s theory of special relativity is based on the assumption (which might be a deep-rooted superstition

More information

Thomson and Rayleigh Scattering

Thomson and Rayleigh Scattering Thomson and Rayleigh Scattering In this and the next several lectures, we re going to explore in more detail some specific radiative processes. The simplest, and the first we ll do, involves scattering.

More information

Thomson and Rayleigh Scattering

Thomson and Rayleigh Scattering Thomson and Rayleigh Scattering Initial questions: What produces the shapes of emission and absorption lines? What information can we get from them regarding the environment or other conditions? In this

More information

Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

More information

Question based on Refraction and Refractive index. Glass Slab, Lateral Shift.

Question based on Refraction and Refractive index. Glass Slab, Lateral Shift. Question based on Refraction and Refractive index. Glass Slab, Lateral Shift. Q.What is refraction of light? What are the laws of refraction? Ans: Deviation of ray of light from its original path when

More information

GRAVITATIONAL FARADAY EFFECT PRODUCED BY A RING LASER

GRAVITATIONAL FARADAY EFFECT PRODUCED BY A RING LASER GRAVITATIONAL FARADAY EFFECT PRODUCED BY A RING LASER DAVID ERIC COX, JAMES G. O BRIEN, RONALD L. MALLETT, AND CHANDRA ROYCHOUDHURI Using the linearized Einstein gravitational field equations and the Maxwell

More information

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

More information

Module 6 : Wave Guides. Lecture 41 : Transverse Electric and Magnetic Mode. Objectives. In this course you will learn the following

Module 6 : Wave Guides. Lecture 41 : Transverse Electric and Magnetic Mode. Objectives. In this course you will learn the following Objectives In this course you will learn the following Important features of Transverse Electric Waves. Fields for Transverse Magnetic (TM) Mode. Important features for Transverse Magnetic (TM) Mode. Important

More information

Mode Patterns of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad

Mode Patterns of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad International Journal of Scientific & Engineering Research Volume 3, Issue 8, August-2012 1 Mode Patterns of Parallel plates &Rectangular wave guides Mr.K.Chandrashekhar, Dr.Girish V Attimarad Abstract-Parallel

More information

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

More information

Acoustic Velocity, Impedance, Reflection, Transmission, Attenuation, and Acoustic Etalons

Acoustic Velocity, Impedance, Reflection, Transmission, Attenuation, and Acoustic Etalons Acoustic Velocity, Impedance, Reflection, Transmission, Attenuation, and Acoustic Etalons Acoustic Velocity The equation of motion in a solid is (1) T = ρ 2 u t 2 (1) where T is the stress tensor, ρ is

More information

08/19/09 PHYSICS 223 Exam-2 NAME Please write down your name also on the back side of this exam

08/19/09 PHYSICS 223 Exam-2 NAME Please write down your name also on the back side of this exam 08/19/09 PHYSICS 3 Exam- NAME Please write down your name also on the back side of this exam 1. A sinusoidal wave of frequency 500 Hz has a speed of 350 m/s. 1A. How far apart (in units of cm) are two

More information

Vector has a magnitude and a direction. Scalar has a magnitude

Vector has a magnitude and a direction. Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude

More information

Sample Questions for the AP Physics 1 Exam

Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each

More information

Astromechanics Two-Body Problem (Cont)

Astromechanics Two-Body Problem (Cont) 5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

More information

Affine Transformations. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Affine Transformations. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Affine Transformations University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.1-5.5. David

More information

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

More information

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at

More information

Q27.1 When a charged particle moves near a bar magnet, the magnetic force on the particle at a certain point depends

Q27.1 When a charged particle moves near a bar magnet, the magnetic force on the particle at a certain point depends Q27.1 When a charged particle moves near a bar magnet, the magnetic force on the particle at a certain point depends A. on the direction of the magnetic field at that point only. B. on the magnetic field

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

DETERMINING THE POLARIZATION STATE OF THE RADIATION CROSSING THROUGH AN ANISOTROPIC POLY (VINYL ALCOHOL) FILM

DETERMINING THE POLARIZATION STATE OF THE RADIATION CROSSING THROUGH AN ANISOTROPIC POLY (VINYL ALCOHOL) FILM DETERMINING THE POLARIZATION STATE OF THE RADIATION CROSSING THROUGH AN ANISOTROPIC POLY (VINYL ALCOHOL) FILM ECATERINA AURICA ANGHELUTA Faculty of Physics,,,Al.I. Cuza University, 11 Carol I Bd., RO-700506,

More information

5.2 Rotational Kinematics, Moment of Inertia

5.2 Rotational Kinematics, Moment of Inertia 5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

More information

Physics in the Laundromat

Physics in the Laundromat Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

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

Basic Electrical Theory

Basic Electrical Theory Basic Electrical Theory Mathematics Review PJM State & Member Training Dept. Objectives By the end of this presentation the Learner should be able to: Use the basics of trigonometry to calculate the different

More information

1 Basic Optics (1.2) Since. ε 0 = 8.854 10 12 C 2 N 1 m 2 and μ 0 = 4π 10 7 Ns 2 C 2 (1.3) Krishna Thyagarajan and Ajoy Ghatak. 1.

1 Basic Optics (1.2) Since. ε 0 = 8.854 10 12 C 2 N 1 m 2 and μ 0 = 4π 10 7 Ns 2 C 2 (1.3) Krishna Thyagarajan and Ajoy Ghatak. 1. 1 1 Basic Optics Krishna Thyagarajan and Ajoy Ghatak 1.1 Introduction This chapter on optics provides the reader with the basic understanding of light rays and light waves, image formation and aberrations,

More information

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

More information

Crystal Optics of Visible Light

Crystal Optics of Visible Light Crystal Optics of Visible Light This can be a very helpful aspect of minerals in understanding the petrographic history of a rock. The manner by which light is transferred through a mineral is a means

More information

DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS

DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS MISN-0-106 DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS Force Center (also the coordinate center) satellite DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS by Peter Signell 1. Introduction..............................................

More information

Solution: (a) For a positively charged particle, the direction of the force is that predicted by the right hand rule. These are:

Solution: (a) For a positively charged particle, the direction of the force is that predicted by the right hand rule. These are: Problem 1. (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields as shown in the figure. (b) Repeat part (a), assuming the moving particle is

More information

State of Stress at Point

State of Stress at Point State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

Motion of Charged Particles in Fields

Motion of Charged Particles in Fields Chapter Motion of Charged Particles in Fields Plasmas are complicated because motions of electrons and ions are determined by the electric and magnetic fields but also change the fields by the currents

More information

LAB 8: Electron Charge-to-Mass Ratio

LAB 8: Electron Charge-to-Mass Ratio Name Date Partner(s) OBJECTIVES LAB 8: Electron Charge-to-Mass Ratio To understand how electric and magnetic fields impact an electron beam To experimentally determine the electron charge-to-mass ratio.

More information

Physics 2B. Lecture 29B

Physics 2B. Lecture 29B Physics 2B Lecture 29B "There is a magnet in your heart that will attract true friends. That magnet is unselfishness, thinking of others first. When you learn to live for others, they will live for you."

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

Waves - Transverse and Longitudinal Waves

Waves - Transverse and Longitudinal Waves Waves - Transverse and Longitudinal Waves wave may be defined as a periodic disturbance in a medium that carries energy from one point to another. ll waves require a source and a medium of propagation.

More information

Reflection of a Gaussian Optical Beam by a Flat Mirror

Reflection of a Gaussian Optical Beam by a Flat Mirror 1 Problem Reflection of a Gaussian Optical Beam by a Flat Mirror Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (October 5, 009) Discuss the flow of energy, including

More information

* Magnetic Scalar Potential * Magnetic Vector Potential. PPT No. 19

* Magnetic Scalar Potential * Magnetic Vector Potential. PPT No. 19 * Magnetic Scalar Potential * Magnetic Vector Potential PPT No. 19 Magnetic Potentials The Magnetic Potential is a method of representing the Magnetic field by using a quantity called Potential instead

More information

PPT No. 26. Uniformly Magnetized Sphere in the External Magnetic Field. Electromagnets

PPT No. 26. Uniformly Magnetized Sphere in the External Magnetic Field. Electromagnets PPT No. 26 Uniformly Magnetized Sphere in the External Magnetic Field Electromagnets Uniformly magnetized sphere in external magnetic field The Topic Uniformly magnetized sphere in external magnetic field,

More information

Ray Optics Minicourse COMSOL Tokyo Conference 2014

Ray Optics Minicourse COMSOL Tokyo Conference 2014 Ray Optics Minicourse COMSOL Tokyo Conference 2014 What is the Ray Optics Module? Add-on to COMSOL Multiphysics Can be combined with any other COMSOL Multiphysics Module Includes one physics interface,

More information

Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

More information

Section 1.7 22 Continued

Section 1.7 22 Continued Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation

More information

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

More information

Binary Stars. Kepler s Laws of Orbital Motion

Binary Stars. Kepler s Laws of Orbital Motion Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

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

Physics of the Atmosphere I

Physics of the Atmosphere I Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de heidelberg.de Last week The conservation of mass implies the continuity equation:

More information

Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop

Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop Vectors and phasors Vectors and Phasors A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles Owen Bishop Copyrught 2007, Owen Bishop 1 page 1 Electronics Circuits

More information

1 of 9 2/9/2010 3:38 PM

1 of 9 2/9/2010 3:38 PM 1 of 9 2/9/2010 3:38 PM Chapter 23 Homework Due: 8:00am on Monday, February 8, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

2D Geometric Transformations. COMP 770 Fall 2011

2D Geometric Transformations. COMP 770 Fall 2011 2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

Coaxial Cable Delay. By: Jacques Audet VE2AZX

Coaxial Cable Delay. By: Jacques Audet VE2AZX Coaxial Cable Delay By: Jacques Audet VE2AZX ve2azx@amsat.org Introduction Last month, I reported the results of measurements on a number of coaxial cables with the VNA (Vector Network Analyzer). (Ref.

More information

Section 10.7 Parametric Equations

Section 10.7 Parametric Equations 299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of

More information

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

More information

Bending of Beams with Unsymmetrical Sections

Bending of Beams with Unsymmetrical Sections Bending of Beams with Unsmmetrical Sections Assume that CZ is a neutral ais. C = centroid of section Hence, if > 0, da has negative stress. From the diagram below, we have: δ = α and s = αρ and δ ε = =

More information

6 J - vector electric current density (A/m2 )

6 J - vector electric current density (A/m2 ) Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems

More information

Notice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)

Notice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2) The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This

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

Refraction of Light at a Plane Surface. Object: To study the refraction of light from water into air, at a plane surface.

Refraction of Light at a Plane Surface. Object: To study the refraction of light from water into air, at a plane surface. Refraction of Light at a Plane Surface Object: To study the refraction of light from water into air, at a plane surface. Apparatus: Refraction tank, 6.3 V power supply. Theory: The travel of light waves

More information

Monochromatic electromagnetic fields with maximum focal energy density

Monochromatic electromagnetic fields with maximum focal energy density Moore et al. Vol. 4, No. 10 /October 007 /J. Opt. Soc. Am. A 3115 Monochromatic electromagnetic fields with maximum focal energy density Nicole J. Moore, 1, * Miguel A. Alonso, 1 and Colin J. R. Sheppard,3

More information

Units and Vectors: Tools for Physics

Units and Vectors: Tools for Physics Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff 1.1.1 The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the

More information

MATH 1231 S2 2010: Calculus. Section 1: Functions of severable variables.

MATH 1231 S2 2010: Calculus. Section 1: Functions of severable variables. MATH 1231 S2 2010: Calculus For use in Dr Chris Tisdell s lectures Section 1: Functions of severable variables. Created and compiled by Chris Tisdell S1: Motivation S2: Function of two variables S3: Visualising

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

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