Physics 1202: Lecture 3 Today s Agenda Announcements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW assignments, solutions etc. Homework #1: On Masterphysics: due this coming Friday Go to the syllabus and click on instructions to register (in textbook section). Make sure to input oyur information to google form https://www.pearsonmylabandmastering.com/no rthamerica/ Labs: Begin this week Today s Topic : Chapter 19: Gauss s law Examples Chapter 20: Electric energy & potential Definition How to compute them Point charges Equipotentials 1
Gauss s Law Gauss s law electric flux through a closed surface is proportional to the charge enclosed by the surface: =4 kq = q " 0 Gauss s Law Useful to get electric field By taking advantage of geometry Charged plate symmetry: E to plate uniformly charged: s = q/a So E: constant magnitude = EA tot =2EA E = 2"0 = q/" 0 = A/" 0 2
Gauss s law Charged line symmetry: E to line uniformly charged: l = q/l So E: constant magnitude DE r E y DE r' Dx A=2pr L x Dx E to end r = EA = E(2 r L) = q/" 0 L E = 2 "0 r = L/" 0 Geometries: Infinite Line of Charge Solution: - symmetry: E x =0 - sum over all elements E x = 0 E y = 1 2λ 4πε 0 r The Electric Field produced by an infinite line of charge is: everywhere perpendicular to the line is proportional to the charge density decreases as 1/r. DE y r Q x l = Q / L : linear charge Dx density r' Dq 3
Geometries: Infinite plane Solution: z - symmetry: E x =E y =0 DE - sum over all elements r' r Q Dq E x = E y = E k =0 E z = E? = 2 0 Dy x Dx s = Q/A : surface charge density y The Electric Field produced by an infinite plane of charge is: everywhere perpendicular to the plane is proportional to the charge density is constant in space! About Two infinite planes? Same charge but opposite Fields of both planes cancel out outside They add up inside E =0 E = 2 0 E = 2 0 2 0 = 0 E = 2 0 - - - - - - - - - - - - - - - - - - - - - - - - - - E =0 Perfect to store energy! 4
Summary Electric Field Distibutions Dipole ~ 1 / r 3 Infinite Plane of Charge Point Charge ~ 1 / r 2 constant Infinite Line of Charge ~ 1 / r 20-1: Electric Potential Q 4pe 0 R Q 4pe 0 r C R R R r r B q B r A A path independence equipotentials 5
Electric potential Energy Recall 1201 E tot = K U kinetic Total mechanical energy Constant for conservative forces potential Potential energy U Depends only on position (ex: U = mgy) Change in U is independent of path U 2, y 2 U 1, y 1 Total energy is Electric potential E ini = K ini U ini and E fin = K fin U fin Total energy is conserved E E fin E ini =0 K K fin K ini U U fin U ini Conservative force E = K U =0 K = U 6
Electric potential Recall from 1201: Work is: W = F Dx But work-energy theorem: W = D K So for conservative forces: D K = -D U W = U By analogy with electric field F ~E ~ Þ U = W q 0 q 0 q 0 SI units: volt () with 1 = 1 J/C Energy Units MKS: U = Q Þ 1 coulomb-volt = 1 joule for particles (e, p,...) 1 e = 1.6x10-19 joules Electrostatic: andegraaff Accelerators electrons 100 ke ( 10 5 e) Electromagnetic: Fermilab protons 1Te ( 10 12 e) 7
E from? We can obtain the electric field E from the potential by inverting our previous relation between E and : Consider 2 plates and a charge q force on q Work done on q ~F = q ~ E W = F x = qe x But work-energy theorem W = K E tot =0= K U F - - - - - - - - - - - - - - - - - - - - - - - - - - Conservative force K = U E from? We can obtain the electric field E from the potential by inverting our previous relation between E and : We have So that W = F x = qe x W = U = q Therefore E = = qe x x F - - - - - - - - - - - - - - - - - - - - - - - - - - 8
We found D = fin - ini. About? Can we define alone? As for gravity, we set a reference point to zero U g = mgh = mg (y fin y ini ) U e = q X X = q ( fin ini ) U g = mgy and U e = q U fin (y fin or fin ) U ini (y ini or ini ) Set to zero 20-2 Motion of Charged Particles in Electric Fields Remember our definition of the Electric Field, F = qe And remembering Physics 1201, F = ma a = q E Now consider particles moving in fields. Note that for a charge moving in a constant field this is just like a particle moving near the earth s surface. a x = 0 a y = constant v x = v ox v y = v oy at x = x o v ox t y = y o v oy t ½ at 2 m 9
Motion of Charged Particles in Electric Fields Consider the following set up, e- - - - - - - - - - - - - - - - - - - - - - - - - - - For an electron beginning at rest at the bottom plate, what will be its speed when it crashes into the top plate? Spacing = 10 cm, E = 100 N/C, e = 1.6 x 10-19 C, m = 9.1 x 10-31 kg Motion of Charged Particles in Electric Fields e- - - - - - - - - - - - - - - - - - - - - - - - - - - v o = 0, y o = 0 v f2 v o2 = 2aDx Or, " v 2 f = 2 qe % $ ' Δx # m & #( v 2 f = 2 1.6x10 19 C) ( 100N /C)& % ( $ % 9.1x10 31 kg ( '( 0.1m) v f =1.9x10 6 m / s 10
Can use energy conservation Recall: E ini = K ini U ini and E fin = K fin U fin Energy conservation: E ini = E fin 1 2 mv2 ini q ini = 1 2 mv2 fin q fin 1 2 m v2 fin vini 2 = q( ini fin )= q r v fin = r v fin = v 2 ini 2q m v 2 ini 2q m E x but E = as before! x 20-3: Point charges Gravitational force ~F g = G m 1m 2 r 2 ˆr Gravitational Potential energy U U = G m 1m 2 r By analogy: ~F e = 1 q 1 q 2 4 0 r 2 ˆr Electric force Þ U e = 1 q 1 q 2 4 0 r Electric potential energy 11
Electric potential Energy Meaning: recall E tot = K U Total energy is conserved ariation of U with r Þ variation of kinetic energy For multiple charges Simple sum Ex: 3 charges U = 1 4 0 q 1 q1 q 2 r 12 q 2q 3 r 23 q 1q 3 r 13 r 13 r 12 q 3 q 2 r 23 Electric Potential By analogy with the electric field Defined using a test charge q 0 ~F = 1 4 0 q 0 Q r 2 ˆr Þ We define a potential due to a charge q ~E = ~ F q 0 = 1 4 0 Q r 2 ˆr Using potential energy of a charge q and a test charge q 0 U = 1 q 0 q 4 0 r = U q 0 = 1 4 0 q r 12
Electric Potential Define the electric potential of a point in space as the potential difference between that point and a reference point. a good reference point is infinity... we typically set = 0 the electric potential is then defined as: for a point charge, the formula is: Lecture 3, ACT 1 A single charge ( Q = -1µC) is fixed at the origin. Define point A at x = 5m and point B at x = 2m. What is the sign of the potential difference between A and B? ( AB º B - A ) -1µC B Á x (a) AB < 0 (b) AB = 0 (c) AB > 0 13
Potential from N charges The potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately. q 1 q 2 r 1 x r 2 r 3 q 3 Þ Electric Dipole z The potential is much easier to calculate than the field since it is an algebraic sum of 2 scalar terms. Rewrite this for special case r>>a: q a a -q q r»r 2 -r 1 r 1 r 2 Þ We can use this potential to calculate the E field of a dipole. Must easier: using E = -D /Dx not here! 14
20-4: Equipotentials We can obtain the electric field E from the potential by inverting our previous relation between E and : We found In general true for all direction E x = E = x,e y = x y,e z = F z - - - - - - - - - - - - - - - - - - - - - - - - - - 20-4: Equipotentials Defined as: The locus of points with the same potential. Example: for a point charge, the equipotentials are spheres centered on the charge. GENERAL PROPERTY: The Electric Field is always perpendicular to an Equipotential Surface. Why?? E x = x,e y = y,e z = z Along the surface, there is NO change in (it s an equipotential!) So, there is NO E component along the surface either E must therefore be normal to surface 15
Equipotential Surfaces: examples For two point charges: 2017 Pearson Education, Inc. Conductors Claim The surface of a conductor is always an equipotential surface (in fact, the entire conductor is an equipotential) Why?? If surface were not equipotential, there would be an Electric Field component parallel to the surface and the charges would move!! Note Positive charges move from regions of higher potential to lower potential (move from high potential energy to lower PE). Equilibrium means charges rearrange so potentials equal. 16
Charge on Conductors? How is charge distributed on the surface of a conductor? KEY: Must produce E=0 inside the conductor and E normal to the surface. Spherical example (with little off-center charge): E=0 inside conducting shell. - - - - - charge density induced on - - inner surface non-uniform. q - - - - - - - - charge density induced on outer surface uniform E outside has spherical symmetry centered on spherical conducting shell. A Point Charge Near Conducting Plane q =0 a - - - - - - -- - -- - - ----- - ---- - - -- - -- - - - - - - 17
A Point Charge Near Conducting Plane q a The magnitude of the force is - F = 1 4πε 2 ( a) 2 0 2 q Image Charge The test charge is attracted to a conducting plane Equipotential Example Field lines more closely spaced near end with most curvature. Field lines ^ to surface near the surface (since surface is equipotential). Equipotentials have similar shape as surface near the surface. Equipotentials will look more circular (spherical) at large r. 18
Equipotential Surfaces & Electric Field An ideal conductor is an equipotential surface If two conductors are at the same potential, the one that is more curved will have a larger electric field around it Think of Gauss s law! This is also true for different parts of the same conductor Explains why more charges at edges Applications: human body There are electric fields inside the human body the body is not a perfect conductor, so there are also potential differences. An electrocardiograph plots the heart s electrical activity An electroencephalograph measures the electrical activity of the brain: 19
Recap of today s lecture Chapter 19: Gauss s law Examples Chapter 20: Electric energy & potential Definition How to compute them Point charges Equipotentials Homework #1 on Mastering Physics From Chapter 19 Due this Friday Labs start this week 20