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Electricity and Magnetism Introduction to Physics 121 • • • • Syllabus, rules, assignments, exams, etc. iClickers Quest Course content overview • Review of vector operations • Dot product, cross product • • • • Scalar and vector fields in math and physics Gravitation as an example of a vector field Gravitational flux, shell theorems, flow fields Methods for calculating fields 1 Copyright R. Janow - Fall 2011 Course Content • • • • • • • • 1 Week: Review of Vectors, Some key field concepts – Prepare for electrostatic and magnetic fields, flux... 5 Weeks: Stationary charges – – Forces, fields, electric flux, Gauss’ Law, potential, potential energy, capacitance 2 Weeks: Moving charges – – Currents, resistance, circuits containing resistance and capacitance, Kirchoff’s Laws, multi-loop circuits 2 Weeks: Magnetic fields (static fields due to moving charges) – – Magnetic force on moving charges, – Magnetic fields caused by currents (Biot-Savart’s and Ampere’e Laws) 2 Weeks: Induction – – Changing magnetic flux (field) produces currents (Faraday’s Law) Thanksgiving in here somewhere 2 Weeks: AC (LCR) circuits, electromagnetic oscillations, resonance Not covered: – Maxwell’s Equations - unity of electromagnetism – Electromagnetic Waves – light, radio, gamma rays,etc – Optics For us, Work units begin with a weekly lecture (on Friday) and ends About a week later when homework is covered in recitation class. There are exceptions due to scheduling. Check Page 3 of the course outline for details. Copyright R. Janow - Fall 2011 Phys 121 – The Big Picture Why do you need Phys 121 – Electromagnetics? → It is fundamental to many areas of Science and Engineering • Electronic circuits (including computers) • Sensors • Biological function • Wireless (and wired) communications Mechanics Phys 111 Electromagnetics Phys 121 “Modern” Physics Phys 234 Typically ECE and Physics majors Copyright R. Janow - Fall 2011 Phys 121 – The Big Picture Capacitors, Resistors, inductors, and Kirchkoff’s loop laws for circuits from Phys 121 are the basics for • Computers Copyright R. Janow - Fall 2011 Phys 121 – The Big Picture Electric fields, voltages, charges from Phys 121 are the basics for medicine and Biology • Electro-cardiography • biological function of Cells Copyright R. Janow - Fall 2011 Phys 121 – The Big Picture Electric fields, voltages, charges from Phys 121 are the basics for Civil Engineering infrastructure • Power Grid • Sensors Copyright R. Janow - Fall 2011 Phys 121 – The Big Picture Electric fields, voltages, charges, circuits from Phys 121 are the basics for Electrical Engineering WiFi 3G, 4G, 5G……. lasers Copyright R. Janow - Fall 2011 Electricity and Magnetism Lecture 01 - Vectors and Fields Physics 121 Review of Vectors : • Components in 2D & 3D. Addition & subtraction • Scalar multiplication, Dot product, vector product Field concepts: • Scalar and vector fields • How to visualize fields: contours & field lines • “Action at a distance” fields – gravitation and electro-magnetics. • Force, acceleration fields, potential energy, gravitational potential • Flux and Gauss’s Law for gravitational field: a surface integral of gravitational field More math: • Calculating fields using superposition and simple integrals • Path/line integral • Spherical coordinates – definition • Example: Finding the Surface Area of a Sphere • Example: field due to an infinite sheet of mass 8 Copyright R. Janow - Fall 2011 Vector Definitions - Experiments tell us which physical quantities are scalars and vectors - E&M uses vectors for fields, vector products for magnetic field and force Representations in 2 Dimensions: y A • Cartesian (x,y) coordinates A A x î A y ĵ A A A 2x A 2y -1 A y tan Ax • Addition and subtraction of vectors: • Notation for vectors: Copyright R. Janow - Fall 2011 ĵ • magnitude & direction k̂ Ay = A sin() î Ax = A cos() x z C A B means Cx A x Bx and Cy A y By C - A means Cx A x and C y A y F ma F ma F ma Vectors in 3 dimensions • Unit vector (Cartesian) notation: • Spherical polar coordinate representation: 1 magnitude and 2 directions Rene Descartes 1596 - 1650 a a x î a y ĵ a zk̂ a (a, , ) • z Conversion into x, y, z components a x a sin cos a y a sin sin a z a cos • a az Conversion from x, y, z components a a 2x a 2y a 2z 1 cos a z / a tan 1 a y / a x ax y x Copyright R. Janow - Fall 2011 ay Definition: Right-Handed Coordinate Systems • • • We always use right-handed coordinate systems. In three-dimensions the righthand rule determines which way the positive axes point. Curl the fingers of your RIGHT HAND so they go from x to y. Your thumb will point in the positive z direction. z y x This course will use many right hand rules related to this one! Copyright R. Janow - Fall 2011 Right Handed Coordinate Systems 1-1: Which of these coordinate systems are right-handed? A. B. C. D. E. I and II. II and III. I, II, and III. I and IV. IV only. y y I. II. x z z x x x III. IV. y z Copyright R. Janow - Fall 2011 z y Vector Multiplication Multiplication by a scalar: sA sA x î sA y ĵ A sA vector times scalar vector whose length is multiplied by the scalar Dot product (or Scalar product or Inner product): B - vector times vector scalar - projection of A on B or B on A A - commutative A B ABcos( ) B A Ax Bx A y By A z Bz Copyright R. Janow - Fall 2011 iˆ ˆj 0, ˆj kˆ 0, ˆi kˆ 0 ˆi ˆi 1, ˆj ˆj 1, kˆ kˆ 1 Vector multiplication, continued Cross product (or Vector product or Outer product): -vector times vector another vector perpendicular to the plane of A and B - draw A & B tail to tail, right hand rule shows direction of C C A B - B A (not commutative) magnitude : C ABsin( ) B A where is the smaller angle from A to B C - If A and B are parallel or the same, A x B = 0 - If A and B are perpendicular, A x B = AB (max) distributiv e rule : A (B Algebra: C) A B A C associative rules : sA (sB) B (sA) B A ( A B) C A (B C) î ĵ k̂, ĵ k̂ î , î k̂ - ĵ Unit vector representation: î î 0, ĵ ĵ 0, k̂ k̂ 0 A B (Ax î Ay ĵ A zk̂) (B x î By ĵ Bzk̂) (A yBz - AzBy ) î (A zBx - AxBz )ĵ (A xBy - AyBx )k̂ Applications: r F L r p F qE Copyright R. Janow - Fall 2011 i j k qv B Example: A force F = -8i + 6j Newtons acts on a particle with position vector r = 3i + 4j meters relative to the coordinate origin. What are a) the torque on the particle about the origin and b) the angle between the directions of r and F. a) Use: r F r F ( 3 î 4 ĵ) ( 8 î 6 ĵ) ( 3 x 8) î î (3 x 6) î ĵ (4 x 8) ĵ î (4 x 6) ĵ ĵ 18 k̂ 32 k̂ b) ˆ 50 k̂ N.m Use: | | r F sin( ) r [ 32 42 ]1 / 2 5 ˆ 50 along z axis F [ 82 62 ]1 / 2 10 r F sin() 50 sin() sin() 1 o 90 that is F r OR Use: | | | r F| r F cos( ) F r 50 cos( ) r F (3x 8) î î (3x 6) î ĵ (4x 8) ĵ î (4x 6) ĵ ĵ 24 24 0 o 50 cos() 0 so 90 that is F r Copyright R. Janow - Fall 2011 Field concepts - mathematical view • • A FIELD assigns a value to every point in space (2D, 3D, 4D) It obeys some mathematical rules: • E.g. superposition, continuity, smooth variation, multiplication,.. • A scalar field maps a vector into a scalar: f: R3->R1 • Temperature, barometric pressure, potential energy • A vector field maps a vector into a vector: f: R3->R3 • Wind velocity, water velocity (flow), acceleration • A vector quantity is assigned to every point in space • Somewhat taxing to the imagination, involved to calculate Example: map of the velocity of westerly winds flowing past mountains FIELD LINES FIELD LINES (streamlines) show wind direction Line spacing shows speed: dense fast Set scale by choosing how many lines to draw Lines begin & end only on sources or sinks Pick single altitudes and make slices to create maps Copyright R. Janow - Fall 2011 ISOBARS EQUIPOTENTIALS Scalar field examples • A scalar field assigns a simple number to be the field value at every point in “space”, as in this temperature map. • Another scalar field: height at points on a mountain. Contours measure constant altitude Contours far apart Contours close together • Contours Grade (or slope or gradient) is related to the horizontal spacing of contours (vector field) flatter steeper Copyright R. Janow - Fall 2011 Side View Slope, Grade, Gradients (another field) and Gravity Height contours h, can also portray potential energy U = mgh. If you move along a contour, your height does not change, so your potential energy does not change. If you move perpendicular to a contour, you are moving along the gradient. • Slope and grade mean the same thing. A 6% grade is a slope of lim h / x dh / dx 0.06 6 x 0 • Gradient is measured along the path. For the case above it would be: 100 lim h / l dh / dl 6/100.2 0.06 x 0 • Gravitational force for example is the gradient of potential energy F dU / dl d (mgh) / dl mg dh / dl dl dh / dl sin() dh F mg sin() • • The steepness and/or force above are related to the GRADIENTS of height and/or gravitational potential energy, respectively, and are also fields. Are the GRADIENTS of scalar fields also scalar fields or are they vector fields? Copyright R. Janow - Fall 2011 Vector Fields • For a vector field the field value at every point in space is a vector – that is, it has both magnitude and direction • A vector field like the altitude gradient can be defined by contours (e.g., lines of constant potential energy – a scalar field). The gradient field lines are perpendicular to the altitude contours • The steeper the gradient (e.g., rate of change of gravitational potential energy) the larger the field magnitude is. DIRECTION • The gradient vectors point along the direction of steepest descent, which is also perpendicular to the contours (lines of constant potential energy). • Imagine rain on the mountain. The vectors are also “streamlines.” Water running down the mountain will follow these streamlines. Copyright R. Janow - Fall 2011 Side View Another scalar field – atmospheric pressure Isobars: lines of constant pressure How do the isobars affect air motion? What is the black arrow showing? Copyright R. Janow - Fall 2011 A related vector field: wind velocity Copyright R. Janow - Fall 2011 Wind speed and direction depend on the pressure gradient Visualizing Fields Could be: • 2 hills, • 2 charges • 2 masses Scalar field: lines of constant field magnitude • • • Altitude / topography – contour map Pressure – isobars, temperature – isotherms Potential energy (gravity, electric) Vector field: field lines show a gradient • • • • Direction shown by TANGENT to field line Magnitude shown by line density - distance between lines Lines start and end on sources and sinks (highs and lows) Forces are fields, but not quite what we call gravitational, electric, or magnetic field Mass or negative charge Examples of scalar and vector fields in mechanics and E&M: TYPE MECHANICS (GRAVITY) FORCE LAW FORCE = SGMm / r2 SCALAR FIELDS GRAV POTENTIAL ENERGY, GRAV POTENTIAL (PE / UNIT MASS) ag = FORCE / UNIT MASS VECTOR =“GRAV. FIELD” FIELDS of GRAVITY Copyright =R.ACCELERATION Janow - Fall 2011 ELECTROSTATICS (CHARGE) COULOMB FORCE Magnetic field around a wire carrying current MAGNETOSTATICS (CURRENT) MAG FORCE = q v X B ELECTRIC POTENTIAL ENERGY ELECTRIC POTENTIAL (PE / UNIT CHARGE) MAGNETIC P. E. OF A CURRENT E = FORCE / UNIT CHARGE = “ELECTRIC FIELD” B = FORCE / CURRENT.LENGTH = “MAGNETIC FIELD” Fields are used to explain “Action at a Distance” (Newton) • A test mass, test charge, or test current placed at some test point in a field feels a force due to the presence of a remote source of field. • The source “alters space” at every test point in its vicinity. • The forces (vectors) at a test point due to multiple sources add up via superposition (the individual field vectors add & cause the force). Field Type Definition Source Acts on Strength (dimensions) gravitational Force per unit mass at test point mass another mass ag = F g / m electrostatic Force per unit charge at test point charge another charge E=F/q magnetic Force per unit current.length electric current another current B Copyright R. Janow - Fall 2011 Gravitation is a Vector Field • • The force of Earth’s gravity points everywhere in the direction of the center of the Earth. The strength of the force is: GMm F 2 r̂ r • • This is an inverse-square force (proportional to the inverse square of the distance). The force is a field mathematically, but it is not quite what we call “gravitational field”. Copyright R. Janow - Fall 2011 m M Idea of a test mass • • The amount of force at some point depends on the mass m at that point GMm F 2 r̂ r What is the force per unit mass? Put a unit test mass m near the Earth, and observing the effect on it: F GM 2 r̂ g(r )r̂ m r • g(r) is the “gravitational field”, or the gravitational acceleration. • The direction (only) is given by r̂ • g(r) vector field, like the force. Copyright R. Janow - Fall 2011 m M Same idea for test charges & currents Meaning of g(r): 1-2: What are the units of: A. B. C. D. E. F GM 2 r̂ g(r )r̂ m r ? Newtons/meter (N/m) Meters per second squared (m/s2) Newtons/kilogram (N/kg) Both B and C Furlongs/fortnight F GM 1-3: Can you suggest another name for 2 r̂ g(r )r̂ ? m r A. B. C. D. E. Gravitational constant Gravitational energy Acceleration of gravity Gravitational potential Force of gravity Copyright R. Janow - Fall 2011 Superposition of fields (gravitational) • • • “Action-at-a-distance”: gravitational field permeates all of space with force/unit mass. “Field lines” show the direction and strength of the field – move a “test mass” around to map it. Field cannot be seen or touched and only affects the masses other than the one that created it. • What if we have several masses? Superposition—just vector sum the individual fields. M • M M M The NET force vectors show the direction and strength of the NET field. The same ideas apply to electric & magnetic fields Copyright R. Janow - Fall 2011 Summarizing: Gravitational field of a point mass M The gravitational field at a point is the acceleration of gravity g (including direction) felt by a test mass at that point • Move test mass m around to map direction & strength of force • Field g = force/unit test mass • Lines show direction of g is radially inward (means gravity is always attractive) • g is large where lines are close together inward force on test mass m gA surfaces of constant field & PE gB rb • Newton: GM g 2 r̂ r rA 2 (Newtons/kg or m/s ) • Field lines END on masses (sources) Where do gravitational field lines BEGIN? • Gravitation is always attractive, lines BEGIN at r = infinity Why inverse-square laws? Why not inverse cube, say? Copyright R. Janow - Fall 2011 M gB gA How long does it take for field to change? Changes in field must propagate from source out to observation point (test mass) at P. For Gravitation, gravity waves For electromagnetism, light waves Action at a distance for E&M travels at speed of light Copyright R. Janow - Fall 2011 An important idea called Flux (symbol F) is basically a vector field magnitude x area - fluid volume or mass flow - gravitational - electric - magnetic Definition: differential amount of flux dFg of field ag crossing vector area dA ag “unit normal” n̂ outward and perpendicular to surface dA dF g flux of a g through dA a g n̂dA (a scalar) Flux through a closed or open surface S: calculate “surface integral” of field over S Evaluate integrand at all points on surface S FS dF a g n̂dA S S EXAMPLE : FLUX THROUGH A CLOSED EMPTY BOX IN A UNIFORM g FIELD • zero mass inside • F from each side = 0 since a.n = 0, F from ends cancels • TOTAL F = 0 • Example could also apply to fluid flow n̂ n̂ n̂ ag What if a mass (flux source) is in the box? Can field be uniform? Can net flux be zero. Copyright R. Janow - Fall 2011 n̂ FLUID FLUX EXAMPLE: WATER FLOWING ALONG A STREAM Assume: • • • • constant mass density constant velocity parallel to banks no turbulence (laminar flow) incompressible fluid – constant r n̂' n̂ A 2 Flux measures the flow (current): • rate of volume flow past a point • rate of mass flow past point • flows mean amount/unit time across area 2 related fields (currents/unit area): • velocity v represents volume flow/unit area/unit time • J = mass flow/unit area/unit time A1 J rv A n̂A n̂ is the outward unit vector to vector area A Flux = amount of field crossing an area per unit time (field x area) V l A l vt The chunk of volume flux v A mass moves l in t t A time t: mass of solid chunk m rV r l A v m l mass flux r A rv A J A and t t Continuity: net flux (fluid flow) through a closed surface = 0 ………unless a source or drain is inside Copyright R. Janow - Fall 2011 Gauss’ Law for gravitational field: The flux through a closed surface S depends only on the enclosed mass (source of field), not on the details of S or anything else GM Gravitational field: g 2 r̂ r 2 (Newtons/kg or m/s ) Consider two closed spherical shells, radii rA & rB centered on M inward force on test mass m surfaces of constant field & PE Find flux through each closed surface F A gA xA A FB gBxAB GM rA2 GM rB2 4GM gA x 4rB2 4GM r x 4rA2 Same! – Flux depends only on the enclosed mass gB rb M gB A FLUX measures the source strength inside of a closed surface - “GAUSS’ LAW” Copyright R. Janow - Fall 2011 gA The Shell Theorem follows from Gauss’s Law 1. The force (field) on a test particle OUTSIDE a uniform SPHERICAL shell of mass is the same as that due to a point mass concentrated at the shell’s mass center (use Gauss’ Law & symmetry) m r x r m x Same for a solid sphere (e.g., Earth, Sun) via nested shells m r r x x + r x + 2. For a test mass INSIDE a uniform SPHERICAL shell of mass m, the shell’s gravitational force (field) is zero • Obvious by symmetry for center point • Elsewhere, integrate over sphere (painful) or apply Gauss’ Law & Symmetry m x x 3. Inside a solid sphere combine the above. The force on a test mass INSIDE depends only on mass closer to the CM than the test mass. x • Example: On surface, measure acceleration distance r from center • Halfway to center, Copyright R. Janow - Fall 2011 ag = g/2 g a 4 Vsphere 3 r 3 When you are solving physics problems, two ways to approach problem Brute force….. Solve equations in 3-D geometry Use intuition to wisely choose a coordinate system and symmetry which help you. How do you choose coordinate system to simplify problem? What direction is x and y direction? Copyright R. Janow - Fall 2011 Symmetry Spherical Symmetry Use Spherical Coordinates Copyright R. Janow - Fall 2011 Symmetry Eg. current in a long, Straight wire Cylindrical Symmetry Use Cylindrical (polar) Coordinates End on view Copyright R. Janow - Fall 2011 Symmetry Planar Symmetry Straight Field lines Use Cartesian Coordinates Copyright R. Janow - Fall 2011 Curved Field lines? WHY? Example: Calculate the field (gravitation) due to a simple source (mass distribution) using superposition Find the field at point P on x-axis due to two identical mass chunks m at +/- y0 y m • Superposition says add fields created at P by each mass chunk (as vectors!!) • Same distances r to P for both masses r0 +y0 r02 x 02 y02 +x0 • Same angles with x-axis cos() x 0 / r0 • Gm x 02 y02 P ag -y0 r0 • Same magnitude ag for each field vector ag ag (from Newtons law of gravitation) m y components of fields at P cancel, x-components reinforce each other a tot a x 2Gm cos() r02 2Gm x 0 r03 where r03 [ x 02 y02 ]3 / 2 • Result simplified because problem has a lot of symmetry Direction: negative x Copyright R. Janow - Fall 2011 x Example: Calculate gravitational field due to mass distributed uniformly along an infinitely long line Find the field at point P on x-axis to y • Similar approach to previous example, but need to include mass from y = – infinity to y =+ infinity • Superposition again: dm = ldy add differential amounts of field created at P by differential mass chunks at y (as vectors!!) r dag y • As before, y components of fields cancel, xcomponents reinforce each other for symmetrically located chunks x P Gdm -y da x dag cos() 2 cos() r l = mass/unit length • Mass per unit length l is uniform, find dm in terms of : x dm ldy lx[1 tan2 ()] d da x Glx[1 tan2 ()] x 2 [1 tan2 ()] cos() d Gl cos() d x • Integrate over from –/2 to +/2 ax Gl / 2 Gl cos( ) d 2 x / 2 x Field of an infinite line falls off as 1/x not 1/x2 Copyright R. Janow - Fall 2011 to y y x tan() dy dtan() x x [1 tan2 ()] d d r 2 x 2 y2 x 2 [1 tan2 ()] /2 cos()d 2 - /2 Line integral (path integral) examples for a gravitational field How much work is done on a test mass as it traverses a particular path dW dU F ds mag ds through a field? B U F ds evaluate along path A Gravitational field is conservative so U is independent of path chosen B A F ds - F ds for any path between A & B A B U F ds 0 for any path closed chosen S circulation,or path integral EXAMPLE uniform field U= - mgh Copyright R. Janow - Fall 2011 U= + mgh test mass Spherical Polar Coordinates in 3 Dimensions (Extra) +z Cartesian r (x, y, z) r x î yĵ zk̂ r z Polar, 3D r (r, , ) rz r cos() " colatitude" , in [0, ] radians " azimuth" , in [0,2 ] radians r (x 2 y 2 z 2 )1 / 2 +y x Polar to Cartesian Copyright R. Janow - Fall 2011 | r | rxy r sin() 90o 90o z r cos() x rxy cos() r sin() cos() y rxy sin() r sin() sin() r2 x 2 y 2 r 2 sin 2 () P 90o y +x Cartesian to Polar cos1 (z / r ) tan-1(y / x) r (x 2 y 2 z2 )1/ 2 Show that the surface area of a sphere = 4R2 (Advanced) READ Copyright R. Janow - Fall 2011 Gravitational field due to an infinite sheet of mass (Advanced) Copyright R. Janow - Fall 2011 Does not depend on distance from plane!