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Transcript
How to Calculate the Electric Field of Objects (Chapter 27) The Electric Field of a Collection of Point Charges: The net electric field due to a group of point charges is where Ei is the field from point charge i. Remember: This is a vector sum! You do not add Magnitudes. E-Field of Two Point Charges The electric field of a point charge q at the origin, r = 0, is where є0 = 8.85 × 10–12 C2/N m2 is the permittivity constant. q1 r1 q2 r2 !" E net = 1 # q1 # q2 # & r + 2 r2 ( 2 1 % 4!" 0 $ r1 r2 ' General Principles of Electric Field Lines • Alternate way to visualize the E-field vector at each point. • E-field vectors are tangent to the field lines at all points. • E-field lines point away from positive charges and towards negative charges. • Density/spacing of field lines=E field magnitude • E-field lines NEVER cross each other!!! (Why ???) Electric field lines: Two Point Charges Compare and contrast dipole and two like charges. Limiting Cases (Important) Limiting cases: A limiting case is the behavior of a physical system when one of the parameters is set to an extreme value. In such cases, complex systems have very simple forms that are easy to analyze mathematically. Example: Localized distribution of charges --->How does it behave at large distances ???? Example: The electric field of 3 point charges (textbook) (Enet )x = (E1 )x + (E2 )x + (E3 )x = 2(E1 )x + (E2 )x E-fields of the charges… 1 q2 (E2 ) x = E2 = 4!" 0 x 2 1 xq (E1 )x = E1 cos# = 4!" 0 (x 2 + d 2 )3/2 (Enet )x = q $ 1 2x ' + 4!" 0 &% x 2 (x 2 + d 2 )3/2 )( As x goes to infinity, the electric field vanishes. (Wrong limit!) Instead let x>>d, 1 3q Enet (x >> d) = 4!" 0 x 2 The limiting case is a point particle. Electric Dipoles • There are two basic types of dipoles – Permanent dipoles such as polar molecules (water) – Induced dipoles, atoms or molecules that are polarized by an electric field • Induced dipoles are important in optics. Optical properties (reflection, refraction, absorption, emission, scattering) of materials due to induced dipoles • Radio antennas are often designed as electric dipoles that oscillate when connected to an AC current producing “dipole radiation”. Polar covalent molecules (eg. water) dipole antenna It’s All About the Dipole Moment We can represent an electric dipole by two opposite charges ±q separated by the small distance 2a. The Dipole-Moment ! p = charge ! displacement ! p = points from negative to positive charge ! p=2qaĵ The electric field can be analyzed as the vector sum of two point charges Calculating the E-field of a Finite Dipole ! " +q !q % E = ke $ 2 r̂+ + 2 r̂! ' r! & # r+ Limit of an “Ideal” Dipole • “Ideal dipole” obtained by taking the limit r>>a • E-field of ideal dipole depends only on p Finite “Real” Dipole Limit of an ideal dipole: Point “Ideal” Dipole Full Expression for Electric Field !" E dipole !" !" 1 3( pir# )r# # p = 3 4!" 0 r !" ( pir# )r# is the component of the dipole moment along the direction of the radial unit vector from the origin The electric field decreases like 1/r3 instead of 1/r2 for a point particle. Why does it fall off faster than that of a point particle ? What kind of force does a dipole feel in an electric field ???? Dipoles in an Electric Field In a constant electric field there is no net force on the dipole. But there is a torque ! Torque is ‘rotational force’: ! ! "! ! =r"F Torque of a Dipole in Constant E Field: Electric Fields Produced by Continuous Charge Distributions: Mathematical Summary 1 ke = 4!" 0 Uniformly Charged Wires and Rods The linear charge density of an object of length L and charge Q, is defined as Electric Field of Finite Length Uniformly Charge Rod/Wire Rod is in x-y plane. By symmetry Ez=0 Solving using Calculus Step 1: General Formula for Charge Distribution Step 2: Express r and dq in terms of y and !. Change integration variable to !. r = y / cos! , dq = " dx = " (dx / d! )d! = d!" y / cos 2 ! using x = y tan ! r! = cos! !j # sin ! !i An Infinite Line of Charge We can obtain the field for an infinite line of charge is x1 and x2 go to inifinity tan !1 = x1 / y tan! 2 = x2 / y lim !1 = $% / 2 lim ! 2 = % / 2 x1 "# x2 "# Inserting into the electric fields… 2k ! 2kQ Ey = k(! / y)(+1 " ("1)) = = y Lr Ex = k(! / y)(0 " 0) = 0 Knight obtains the same answer by different means… Where r is the radial distance away from the rod. Limiting Cases: Electric Field of Point Charge, Finite Line Segment, and Infinite Line Charge Very close to the line segment, it’s E-Field is indistinguishable from infinite line charge Very far away, the line segment looks like a point charge A Ring of Charge P is on the axis of the ring at a distance x from center. Want to find E-field at P Setting up the problem… Because of the symmetry with respect to rotations around the ring axis, the field perpendicular to axis vanishes, E! = 0 dq = ! dl = ! (ad" ) r= a +x 2 2 Adding up the E-field… Step 1: E-field due to single charge element dq Step 2: Integrate dE with respect to dq around the ring. Because of the symmetry of P on the ring axis, x and r do not depend on the position of the charge element dq. Finalize the solution (ring of charge) Step 3: Note that the integral of dq around the ring is the total charge of the ring, Q Step 4: Put all of the pieces together What is the limiting case ?? A Disk of Charge We can obtain the E-field for a disk by dividing the disk into infinitesimal rings of charge and use our result for a ring of charge. Q Q != = 2 A "R Calculating the E-field for a Disk Step 1: A ring of radius r has a thickness dr and charge dQ = 2!"rdr Step 2: Electric field due to the ring on axis is dEz = k (z zdQ 2 +r ) 2 3/2 Step 3: Integrate the E-fields for all ring segments comprising the disk R R Ez = ! dEz = ! 0 0 (z kz 2 +r ) 2 3/2 2"#r dr % ( z Ez = 2" k# ' 1 $ 2 2 * & z +R ) E-field for in Infinite Sheet of Charge If we let the size of the disk become infinite, we obtain the E-field for an infinite plane of charge ( Edisk )z & ) z " = 2! k" lim ( 1 % = 2! k" = 2 2 + R#$ 2, 0 ' z +R * E field does not decrease with distance ! How can this be a correct limiting case ??? A Plane of Charge ">0, E points away from charge sheet. "<0, E points toward the charge sheet. E-field dependence on distance (x) for finite disk, infinite sheet, and point charge Disk size=R=5cm Summary of different charge distribution: Dependence on distance r • • • • Electric Dipole: E-field falls off like 1/r3 Point Charge: E-field falls off like 1/r2 Infinite line of charge: E-field falls off like 1/r Infinite plane/sheet of charge: E-field is constant ! The Parallel-Plate Capacitor Two electrodes, one with charge +Q and the other with –Q placed face-toface a distance d apart. Capacitors: Critical to Technology • Capacitors are one type of passive components used in electrical circuits. Other passive components are resistors, inductors, and diodes. Unlike active components (transistors, op-amps, logic gates), passive components have a single input and output lead. • Capacitors store electrical energy that can later be released very quickly • Capacitors are essential elements along with resistors and inductors for constructing analog filters and electrical oscillators. Uses for Capacitors • • • • • • • • • • Temporary energy storage (eg. to prevent loss of data when changing batteries of a device) Pulsed power: Store and release energy in pulses to power devices like high power pulsed lasers ,camera flashes. Computer memory: DRAM memory stores each bit in the form of charge on a capacitor Digital Cameras: CCD image sensors use a capacitor array to store the captured image as electrical charges. Power conditioning: Capacitors are used to smooth out current fluctuations from AC wall current to provide smooth steady currents for other circuits (such as audio equipment, computers) Signal processing: Capacitors are essential for analog filters used to remove high frequency noise from a AC signal, block DC components of AC signals, or as radio tuners Microphones (Condenser microphones are a variable capacitor) Fingerprint sensors Airbag sensors rely on a mechanical variable capacitor (MEMS). The iPhone touch screen is a pressure sensitive capacitor. Condenser Microphone iPhone Touch Screen But it doesn’t look like a parallel plate capacitor… An early parallel plate capacitor from the 1800’s Capacitors in modern circuits don’t look like the parallel plates in the textbook. Why ? Types of Capacitors Parallel-Plate Capacitor Cylindrical Capacitor A cylindrical capacitor is a parallel-plate capacitor that has been rolled up with an insulating layer between the plates. L d The ideal capacitor assumes that the plates are infinite in size. Real capacitors have finite plates and non-uniform‘fringe fields’. When can we approximate a real capacitor to be ideal ??? Deriving the electric field Ecapacitor ! Q = = "0 "0 A Motion of a Charged Particle in an Electric Field The electric field exerts a force on a charged particle. If this is the only force acting on q, it causes the charged particle to accelerate with Two particles with the same q/m have the exact same trajectories! In a uniform field, the acceleration is constant: ! d x q ! = E 2 dt m 2 1 " q !% 2 ! ! ! ! x(t) = $ E ' t + v(0)t + x(0) 2# m &