#### Transcript Lecture01: Introduction, Vectors, Scalar and Vector Fields

Physics 121: Electricity and Magnetism Introduction Syllabus, rules, assignments, exams, etc. Text: Young & Friedman, University Physics Homework & Tutorial System: Mastering Physics Course Content: • • • • • • 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, RC 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 & Inductance – Changing magnetic flux (field) produces currents (Faraday’s Law) – Inductance, LR Circuits 2 – 3 Weeks: AC (LCR) circuits, – Eelectromagnetic oscillations, Resonance – Impedance, Phasors Not covered: – Maxwell’s Equations - unity of electromagnetism – Electromagnetic Waves – light, radio, gamma rays,etc 1 Copyright R. Janow – Spring 2016 Physics 121 - Electricity and Magnetism Lecture 01 - Vectors and Fields Review of Vectors : • Components in 2D & 3D. Addition & subtraction • Scalar multiplication, Dot product, Vector product Field concepts: • Scalar and vector fields in math & physics • 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 integral/line integral • Spherical coordinates – definition • Example: Finding the Surface Area of a Sphere • Example: field due to an infinite sheet of mass 2 Copyright R. Janow – Spring 2016 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 uses several right hand rules related to this one! Copyright R. Janow – Spring 2016 Cartesian & Spherical Polar Coordinates – 3D +z Cartesian r (x, y, z) r x î yĵ zk̂ r z Polar, 3D r (r, q, f) rz r cos(q) r (x 2 y 2 z2 )1 / 2 q q " colatitude" , in [0, ] radians f " azimuth" , in [0,2 ] radians +y x z r cos(q) r2 x2 y 2 r 2 sin2 (q) x r cos(f) r sin(q) cos(f) y r sin(f) r sin(q) sin(f) Copyright R. Janow – Spring 2016 | r | rxy r sin( q) f 90o 90o Polar to Cartesian P 90o y +x Cartesian to Polar q cos1 (z / r ) f tan-1(y / x) r (x 2 y 2 z2 )1/ 2 Surface Integral Example: Show that the surface area of a sphere A= 4R2 by integrating over the sphere’s surface Find dA – an area segment on the surface of the sphere, then integrate on angles f (azimuth) and q (co-latitude). z dA dl dh dA r̂ dA Where: • dl is a curved length segment of the circle around the z-axis (along a constant latitude line) • dh is a segment along the q direction (along a constant longitude line) dl r sin(q) df q r f y dh r dq x q [0, ] Angle range for a full sphere: A r̂.dA surface f [0, 2 ] Factors into 2 simple 1 dimensional integrations 2 0 0 2 0 0 r sin(q) dqdf r { df} { sin(q) dq } 2 2 2r 2 sin(q) dq 2r 2 [ cos(q)] 0 2r 2 ()[ 1 1] 0 A 4 r 2 Copyright R. Janow – Spring 2016 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 z y Ans: D Copyright R. Janow – Spring 2016 There are 3 Kinds of Vector Multiplication Multiplication of a vector 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 f A - vector times vector scalar - projection of A on B or B on A - commutative AoB ABcos( ) BoA A xBx A yB y A zBz unit vectors measure perpendicularity: Copyright R. Janow – Spring 2016 î ĵ 0, ĵ k̂ 0, î k̂ 0 î î 1, ĵ ĵ 1, k̂ k̂ 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( f) B A where f is the smaller angle from A to B f 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 xB y - AyBx )k̂ Applications: r F L r p F qE Copyright R. Janow – Spring 2016 i j k qv B What’s a “Field” - Mathematical View • • • • A FIELD assigns a value to every point in space (2D, 3D, 4D,….) It may have nice mathematical properties, like other functions: E.g. superposition, continuity, smooth variation, multiplication,.. A scalar field f maps a vector into a scalar: f: R3->R1, f: R2->R1 … • A scalar quantity is assigned to every point in 3D space ISOBARS • Temperature, barometric pressure, potential energy EQUIPOTENTIALS • A vector field g maps a vector into a vector: g: R3->R3, g: R2->R2 • A 2D/3D vector is assigned to every point in 2D/3D space • Wind velocity, water velocity (flow), acceleration • Taxing to the imagination, involved to calculate Example: map of the velocity of westerly winds flowing past mountains Pick single altitudes and make slices to create maps Copyright R. Janow – Spring 2016 … 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 Scalar field examples A scalar field assigns a simple number as the field value at every point in “space”. • • • Altitude map shows heights of points on a mountain as function of x-y position. All points on a contours have the same altitude • Temperature map portrays ground-level temperature as function of x-y position Maps R2 -> R1 Contours far apart Contours closely spaced • Contours Grade (or slope) is related to the horizontal spacing of contours (vector field) flatter steeper Copyright R. Janow – Spring 2016 Side View Vector Fields • The value of a vector field at every point in space is a vector – with magnitude and direction • A vector field (e.g.,gravitational force) can be generated by taking the gradient of a scalar field (e.g.,potential energy). • Gradient field lines are perpendicular to the contours (e.g., lines of constant potential energy) • The steeper the gradient (e.g., rate of change of gravitational potential energy) the larger the field magnitude is. DIRECTION • Gradient vectors point along the direction of steepest descent, which is also perpendicular to the contours. • Imagine rain flowing down a mountain. The vectors are also “streamlines.” Water running down the mountain will follow these streamlines. Copyright R. Janow – Spring 2016 Side View Slope, Grade, Gradients (another field) and Gravity Height contours portray constant potential energy U = mgh. Motion perpendicular to a contour at a point is along the gradient. • Slope and grade mean the same thing. A 15% grade is a slope of 15 lim h / x dh / dx 0.15 x 0 • 100 Gradient is measured along the path. For the case at right it would be: lim h / l dh / dl 15 /101.1 0.148 x 0 • 15% Gravitational force component along path l is the gradient of potential energy l F dU / dl d (mgh) / dl mg dh / dl h dh / dl sin(q) F mg sin(q) • • q x The GRADIENT of height (or gravitational potential energy) is also a field representing steepness (or force) Are the GRADIENTS of scalar fields also scalar fields or are they vector fields? Copyright R. Janow – Spring 2016 Another scalar field – atmospheric pressure Isobars: lines of constant pressure How do the isobars affect air motion? What are the black arrows showing? Copyright R. Janow – Spring 2016 A related vector field: wind velocity Wind speed and direction depend on the pressure gradient Copyright R. Janow – Spring 2016 “Action at a Distance” forces are also called fields • Place a test mass, test charge, or test current at some test point in a field • It feels a force due to the presence of remote sources of the field. • The sources “alter space” at every possible test point. • The forces (vectors) at a test point due to multiple sources add up via superposition (individual field vectors add & form the net field). Field Type Source Acts on Definition Strength (dimensions) mass another mass Force per unit mass at test point ag = F g / m electrostatic charge another charge Force per unit charge at test point E=F/q magnetic electric current .length another current .length Force per unit current.length B ~ F/qv or F/iL gravitational Copyright R. Janow – Spring 2016 Summary: Visualizing Physical 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 proportional to line density inversely to distance between lines Lines start and end on sources and sinks of field (highs and lows) Forces are fields, with direction related to gravitational, electric, or magnetic field Mass or negative charge Magnetic field around a wire carrying current Summary: Scalar and vector fields in mechanics and E&M: TYPE MECHANICS (GRAVITY) ELECTROSTATICS (CHARGE) FORCE (vector) Gravitational Force = GMm / r2 Coulomb Force = kqQ / r2 SCALAR FIELDS Gravitational Potential Energy Gravitational Potential (PE / UNIT MASS) Electric Potential Energy Electric Potential (volts) (PE / UNIT CHARGE) Magnetic P. E. (due to a current) E = Force / Unit Charge = “Electric Field” B = Force / Unit Current x Length = “Magnetic Field” ag = Force / Unit Mass VECTOR = “Gravitational Field” FIELDS = Acceleration of Gravity “g” Copyright R. Janow – Spring 2016 MAGNETOSTATICS (CURRENT) Magnetic Force = q v X B Gravitational field of a point mass M Point mass Spherical symmtery 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 surfaces of constant field & PE inward force on test mass m gA gB rb • From Newton force law: GM g 2 r̂ r 2 (Newtons/k g or m/s ) rA M gB • 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? The same ideas apply to electric fields Copyright R. Janow – Spring 2016 gA 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 fields Copyright R. Janow – Spring 2016 An important idea called Flux (symbol F): 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) “Phi” 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, RECTANGULAR 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 – Spring 2016 n̂ FLUID FLUX EXAMPLE: WATER FLOWING ALONG A STREAM Self Study Assume: • • • • constant mass density r incompressible fluid constant flow velocity parallel to banks no turbulence (laminar flow) n̂' n̂ A 2 Flux measures the flow (current): • flow means amount/unit time across area • either rate of volume flow past a point …or… rate of mass flow past point A 1 Two related fluid flow fields (currents/unit area) appear: • velocity v represents volume flow/unit area/unit time • J = mass flow/unit area/unit time A n̂A n̂ is the outward unit vector to vector area A J rv 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 fluid moves L t t A in time t: r r mass of solid chunk m V L A v m L and mass flux r r A v A J A t t Continuity: Net flux (fluid flow) through a closed surface = 0 ………unless a source or drain is inside Copyright R. Janow – Spring 2016 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 Example: spherically symmetric mass distribution, radial gravitational field GM Field: g 2 r̂ r 2 (Newtons/k g or m/s ) GM d(flux ) f ield . dA g . dA 2 r̂.dA r Find total flux through closed surface A GM GM F A g.dA 2 r̂.dA 2 r̂.dA A A A rA r inward acceleration of test mass m Spherical surface of constant field & PE gA Integral for surface area of sphere GM F A g.dA 2 x 4 rA2 4GM A rA rA M Flux depends only on the enclosed mass (same flux for any closed surface enclosing M) FLUX measures the strength of a field source that is inside a closed surface - “GAUSS’ LAW” Copyright R. Janow – Spring 2016 gA 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 or see section 13.6) m r r m x 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 m x x • Obvious by symmetry for center point • Elsewhere, integrate over sphere (painful) or apply Gauss’ Law & Symmetry to a concentric sphere inside the shell 3. Inside a solid sphere of mass combine above. 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 – Spring 2016 ag = g/2 g a 4 3 Vsphere r 3 Superposition Example: Calculate the field (gravitational) due to two point masses at a special point Find the field at point P on x-axis due to two identical mass chunks m at +/- y0 • Superposition says add fields created at P by each mass chunk (as vectors!!) • Same distances r0 to P for both masses y m r0 +y0 r02 x 02 y02 q +x0 • Same angles with x-axis cos(q) x 0 / r0 • Gm x 02 y02 q P ag -y0 r0 • Same magnitude ag for each field vector ag ag (from Newtons law of gravitatio n) m y components of fields at P cancel, x-components reinforce each other a tot a x 2Gm cos(q) r02 2Gm x 0 r03 where • Result simplified because problem had a lot of symmetry Copyright R. Janow – Spring 2016 r03 [ x 02 y02 ]3 / 2 Direction: negative x x Example: Calculate gravitational field due to an infinitely long line of uniformly distributed mass. Find the field at point P on x-axis to y • Integrate over the source of field holding P fixed • Add differential amounts of field created at P by differential point mass chunks at y (vectors!!) dm = ldy • Include mass from y = – infinity to y =+ infinity • For symmetrically located point mass chunks dm: r y • y-components of fields cancel, • x-components of fields reinforce • Mass per unit length l is uniform, find dm in terms of q: -y a x da x where Gdm da da cos( q ) cos(q) - x g r2 dm ldy lx[1 tan2 (q)] dq da x Glx[1 tan2 (q)] cos(q) dq x 2 [1 tan2 (q)] • Integrate over q from –/2 to +/2 ax Gl cos(q) dq x Gl / 2 Gl cos( q ) d q 2 x / 2 x Field of an infinite line falls off as 1/x not 1/x2 Copyright R. Janow – Spring 2016 x CYLINDRICAL SYMMETRY dag q q P x l = mass/unit length to y y x tan(q) dy dtan(q) x x [1 tan2 (q)] dq dq r 2 x 2 y2 x 2 [1 tan2 (q)] /2 cos(q)dq 2 - /2 Example of a “line integral” (path integral)- Work done on a mass traversing a gravitational field Self Study 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 B U F ds evaluate along path test mass A Gravitational field is conservative, meaning U is independent of path chosen A F d s F d s B A for any path between A & B B U F ds 0 for any closed path that is chosen S circulation,or path integral EXAMPLE uniform field U= - mgh Copyright R. Janow – Spring 2016 U= + mgh Vectors in 3 dimensions • Cartesion unit vector representation: • Spherical polar coordinate representation: 1 magnitude and 2 directions Rene Descartes 1596 - 1650 a a x î a y ĵ a zk̂ a (a, q, f) • z Conversion into x, y, z components a x a sin q cos f a y a sin q sin f a z a cos q • a az Conversion from x, y, z components q a a 2x a 2y a 2z 1 ax q cos a z / a f tan 1 a y / a x x Copyright R. Janow – Spring 2016 ay f a sin(q) y Gravitational field due to an infinite sheet of mass Simple 2 dimensional Copyright R. Janow – Spring 2016 integration Self Study Constant - does not depend on distance from plane!