Physics 121: Electricity and Magnetism Introduction

1. Physics 121: Electricity and Magnetism Introduction • Syllabus, rules, assignments, exams, etc. • Text: Young & Friedman, University Physics, 13th Edition • Homework & Tutorial System: Mastering Physics • 5 Weeks: Stationary charges – • 2 Weeks: Moving charges – • 2 Weeks: Magnetic fields (static fields due to moving charges) • 2 Weeks: Induction & Inductance • 2 – 3 Weeks: AC (LCR) circuits, Overview: What do we cover?: Physics 121 viewed at 100,000 feet: • Nature seems to have 4 basic forces - Gravitation, electromagnetic, nuclear, weak interaction • E&M is more abstract and conceptual than mechanics • Fields: You can’t see fields or directly feel them • Many E&M topics are inherently 3D • You need to develop thinking in 3D through practice • Calculus including integration will be lightly used at first (ouch!) • Start simple, then surface, path, multiple integrals, some series. • Be sure you’re taking or took Calc II • Symmetry as simplifier

2. Physics 121 - Electricity and MagnetismLecture 01 - Vectors and Fields • Review of Vectors & 3D Coordinates : • Right Handed Coordinates • Spherical and Cartesian Coordinates in 3D. • Scalar multiplication, Dot product, Vector product • Example: find the surface area of a Sphere • 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 • More math (for Reference): • Flux and Gauss’s Law for gravitational field: a surface integral of gravitational field • Calculating fields using superposition and basic integrals • Path integral/line integral • Example: field due to an infinite sheet of mass

3. y Ay = A sin(q) A • Cartesian (x,y)coordinates q Vectors in 2 Dimensions Ax = A cos(q) x • Magnitude &direction z • Addition and subtraction • of vectors: • Notation for vectors: Vectors & Scalars - The world has 3 spatial dimensions (plus time) - Experiments tell us which physical quantities are scalars and vectors - E&M uses vector and scalar fields - E&M requires vector multiplication for magnetic field and force - E&M requires 3 dimensional treatment (especially magnetism)

4. z y x Vectors in Three DimensionsDefinition: Right-Handed Coordinate Systems • We always use right-handed coordinate systems. • In three-dimensions the right-hand 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. • Do 2 dimensional axes have different “handedness”? This course uses several right hand rules related to this one!

5. Three Dimensions: Cartesian & Spherical Polar Coordinates +z Cartesian P 90o Polar, 3D q Polar to Cartesian +y Cartesian to Polar f 90o 90o +x Dimensionality: How many independent pieces of information does it take?

6. y y II. I. z x x z x x III. IV. y z z y Right Handed Coordinate Systems 1-1: Which of these coordinate systems are right-handed? • I and II. • II and III. • I, II, and III. • I and IV. • IV only. Ans: D

7. Multiplication of a vector by a scalar: vector times scalar  vector whose length is multiplied by the scalar direction unchanged Dot product (or Scalar product or Inner product): - vector times vector  scalar - uses projection of A on B or B on A - commutative unit vector dot products measure perpendicularity: f Manipulating Vectors: 3 Kinds of Vector Multiplication

8. 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 Ø f • If A and B are parallel or the same, A x B = 0 • If A and B are perpendicular, A x B = AB (max) Algebra: Unit vector representation: i j k Applications: Vector multiplication, continued

9. z • dh is a curved length segment along the q direction • (constant longitude line) • dl is a curved length segment of the circle around • the z-axis (constant latitude line) q r y f x Angle ranges for full sphere: Integral for area A factors into 2 simple 1 dimensional integrals USED IN MANY EXAMPLES SURFACE INTEGRAL EXAMPLE: Show that the surface area of a sphere is A= 4pR2: Integrate over the surface using 3D spherical coordinates SPHERICAL SYMMETRY - All points have same r. Angles q and f vary. Find expression for an area segment dA on the surface of the sphere, then integrate on angles f (azimuth) and q (co-latitude).

10. 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 (fluid flow), acceleration • Taxing to the imagination, involved to calculate FIELD LINES Example: map of the velocity of westerly winds flowing past mountains (vector field) “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 v: R3->R3 Pick single altitudes and make slices to create maps What’s a “Field” - Mathematical View • A FIELD assigns a value to every point in space (2D, 3D, 4D, domain) • It has nice mathematical properties, like other functions: e.g. superposition, continuity, smooth variation, associative 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 • Temperature, barometric pressure, potential energy, altitude map, ISOBARS EQUIPOTENTIALS

11. Altitude map shows heights of points on a mountain as function of x-y position. • All points on a contour have the same altitude Contours far apart Contours closely spaced Contours • Grade (or slope) is related to the horizontal spacing of contours (vector field) flatter steeper Side View Scalar field examples A scalar field assigns a simple number as the field value at every point in “space”. Temperature map portrays ground-level temperature as function of x-y position. Maps R2 -> R1

12. Side View Vector field examples A vector field assigns a vector value (magnitude and direction to every point in space. • The gradient of a scalar field (such as gravitational potential energy) is a vector field (e.g.,gravitational force component along surface). • Gradient field lines are perpendicular to the scalar field contours (e.g., lines of constant potential energy) • The steeper the gradient (such as the rate of change of gravitational potential energy) the larger the vector field magnitude is (i.e., the force). FIELD DIRECTION • Gradient vectors point along the direction of steepest descent (fastest change) of the scalar field, and are also perpendicular to the contours. • Imagine rain flowing down a mountain. The gradient vectors are also “streamlines.” Water running down the mountain will follow these streamlines.

13. “Action at a Distance” forces are represented by fields • Gravitation Example: • Place a test mass at some test point in a field • The force on it is due to the presence of remote masses (field sources) that • “alter space” at every possible test point. • The net force (vector) at each test point is due to multiple masses whose • effects add up via superposition (individual field vectors add up). • Charges and current lengths (for electrostatic & magnetic fields) act similarly.

14. surfaces of constant field & PE inward force on test mass m • Move test mass m around to map direction • & strength of force • Field g = force/unit test mass • Field lines show direction of g is radially • inward (means gravity is always attractive) • g is large where lines are close together gB gA rB • From Newton force law: gA gB rA M • Field lines END on masses (sources) Gravitational field of a point mass M Point mass -- Spherical symmetry The gravitational field at a point is the acceleration of gravity g (including direction) felt by a test mass m at that point • 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

15. M M M M • The NET force vectors show the direction and strength of the NET field. Fields and Superposition (Gravitation example) • “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 is effect of several masses? Superposition—just vector sum the individual fields. The same ideas apply to electric fields

16. y Dm r0 ag +y0 q +x0 q x P ag -y0 r0 Dm Direction: negative x Superposition Example: Field (gravitational) due to two point masses at a special point “P” (symmetric) • Find the field at point P on x-axis due to • identical mass chunks Dm at +/- y0 • Superpositionsays add fields created at P • by each mass chunk (as vectors!!) • Same distances r0 to P for both masses • Same angles q with x-axis • Same magnitude agfor each field vector • y components of fields at P cancel by symmetry, x-components reinforce each other • Calculation simplified because problem had a lot of symmetry

17. Supplementary Material

18. Grade means the same thing as slope. A 15% grade is a slope of • Gradient is measured along the path. For the case at left it would be: 15 15% • The gradient of the potential energy is the gravitational force component along path Dl: 100 Dl Dh Dx q Distinction: Slope, Grade, Gradients in Gravitational Field Height contours portray constant gravitational potential energy DU = mgDh. Force is along the gradient, perpendicular to a potential energy contour. • The GRADIENT of height (or gravitational potential energy) is a vector field representing steepness (or gravitational force) • In General: The GRADIENTS of scalar fields are vector fields.

19. Could be: • 2 hills, • 2 charges • 2 masses Vector field: field lines point along gradient of a scalar field • 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 (with some exceptions like magnetic field) • Forces are fields, with direction related to gravitational, electric, or magnetic field Magnetic field around a wire carrying current Mass or negative charge Summary: Visualizing Physical Fields Scalar field: use lines of constant field magnitude • Altitude / topography – contour map • Pressure – isobars, temperature – isotherms • Potential energy (gravity, electric) • Electric potential, Electric Potential Energy

20. v “unit normal” outward and perpendicular to surface dA • EXAMPLE : FLUID FLUX THROUGH A CLOSED, EMPTY, RECTANGULAR BOX IN A UNIFORM VELOCITY FIELD • No sources or sinks of fluid inside • DF from each side = 0 since v.n = 0, DF from ends cancels • TOTAL F = 0 • Example could also apply to gravitational or electric field What if a flux source or sink (or mass, or charge) is in the box? Can field be uniform? Can net flux be zero. v An important idea called Flux (symbol F): Basically a vector field magnitude x area In a fluid, flux measures volume flow or mass flow per second. Definition: differential amount of flux dF of field vcrossing vector area dA “Phi” Flux through a closed or open surface S: calculate “surface integral” of field over S Evaluate integrand at all points on surface S

21. Two related fluid flow fields (currents/unit area): • Velocity field v is volume flow across area/unit time • Mass flow field J is mass flow across area/unit time Flux = amount of field crossing an area per unit time (fieldo area) The chunk of fluid moves distance DL in time Dt: r r º D = r D = r D D m V L A o mass of solid chunk r r r r r D D m L r \ º = r D = r D = D mass f lux A v A J A and o o o D D t t MORE ON FLUID FLUX: WATER FLOWING ALONG A STREAM • Assume: • fluid with constant mass density (incompressible) • constant flow velocity parallel to banks • no turbulence (smooth laminar flow) • Flux measures the flow (current): • flow means quantity/unit time crossing area • either rate of volume flow past a point …or… • rate of mass flow past point Continuity: Net flux (fluid flow) through a closed surface = 0 ………unless a source or sink is inside

22. Spherical Surface S of constant field & PE inward acceleration of test mass m g g r M Flux depends only on the enclosed mass (Actually, same flux for any closed surface enclosing M !!) FLUX measures the strength of a field source that is inside a closed surface - “GAUSS’ LAW” Gauss’ Law for gravitational field: The flux of gravitational field 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 M, radial gravitational field Field: Find total flux through closed surface S Integral for surface area of sphere

23. Gauss’ Law implies Shell Theorem for Gravitation m r r m x x m r r r + + x x x 2. For a test mass INSIDEa uniform SPHERICAL shell of mass m, the shell’s gravitational force (field) is zero m • Obvious by symmetry at the center point • Elsewhere, integrate over sphere (painful) or apply Gauss’ • Law & Symmetry to a concentric spherical Gaussian • surface inside the shell x x 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. • Example: On surface, measure acceleration g a • distance r from center • Halfway to center, ag = g/2 x 1. The force (field) on a test particle OUTSIDE a UNIFORMSPHERICALSHELL 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) Same for a solid sphere (e.g., Earth, Sun) via nested shells

24. dm = ldy r dag y q x q x P where -y l = mass/unit length • Integrate over q from –p/2 to +p/2 Field of an infinite line falls off as 1/x not 1/x2 Example: Calculate gravitational field due to an infinitely long line of uniformly distributed mass on y-axis. Find the field at point P on x-axis • Integrate over the source of field. Hold P fixed • Add differential amountsof field createdat Pby • differential point mass chunksat y(vectors!!) • Include mass from y = – infinity to y =+ infinity • For symmetrically located point mass pairs dm: • y-components of fields cancel, • x-components of fields reinforce • Mass per unit length l is uniform, find dm in terms of q: CYLINDRICAL SYMMETRY

25. z P a r y f a dm x • By symmetry, g is along z axis, so take z component Gravitational field due to an infinite sheet of mass in x-y plane • s = mass/unit area in x-y plane, uniform • Find field at point P on z-axis. • dg is field magnitude due to point mass dm. • Represent dm as sector of width adf and radius da • Rewrite da in terms of s, a, f, & z

26. Integrations factor. Trivial f integration is 2p. Simple a integration. • Field is constant – does not depend on distance from the plane • Depends only on surface mass density • Points toward plane. Gravitational field… infinite sheet of mass… continued • After cancellation in above: • Integrate over the plane by doing two-fold integration on angles.