The Electric Field Driven Generator
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The Lorentz Force
CLASSICAL CONCEPT REVIEW 14 The Lorentz Force We can find empirically that a particle with mass m and electric charge q in an elec- tric field E experiences a force FE given by FE = q E LF-1 It is apparent from Equation LF-1 that, if q is a positive charge (e.g., a proton), FE is parallel to, that is, in the direction of E and if q is a negative charge (e.g., an electron), FE is antiparallel to, that is, opposite to the direction of E (see Figure LF-1). A posi- tive charge moving parallel to E or a negative charge moving antiparallel to E is, in the absence of other forces of significance, accelerated according to Newton’s second law: q F q E m a a E LF-2 E = = 1 = m Equation LF-2 is, of course, not relativistically correct. The relativistically correct force is given by d g mu u2 -3 2 du u2 -3 2 FE = q E = = m 1 - = m 1 - a LF-3 dt c2 > dt c2 > 1 2 a b a b 3 Classically, for example, suppose a proton initially moving at v0 = 10 m s enters a region of uniform electric field of magnitude E = 500 V m antiparallel to the direction of E (see Figure LF-2a). How far does it travel before coming (instanta> - neously) to rest? From Equation LF-2 the acceleration slowing the proton> is q 1.60 * 10-19 C 500 V m a = - E = - = -4.79 * 1010 m s2 m 1.67 * 10-27 kg 1 2 1 > 2 E > The distance Dx traveled by the proton until it comes to rest with vf 0 is given by FE • –q +q • FE 2 2 3 2 vf - v0 0 - 10 m s Dx = = 2a 2 4.79 1010 m s2 - 1* > 2 1 > 2 Dx 1.04 10-5 m 1.04 10-3 cm Ϸ 0.01 mm = * = * LF-1 A positively charged particle in an electric field experiences a If the same proton is injected into the field perpendicular to E (or at some angle force in the direction of the field. -
Capacitance and Dielectrics Capacitance
Capacitance and Dielectrics Capacitance General Definition: C === q /V Special case for parallel plates: εεε A C === 0 d Potential Energy • I must do work to charge up a capacitor. • This energy is stored in the form of electric potential energy. Q2 • We showed that this is U === 2C • Then we saw that this energy is stored in the electric field, with a volume energy density 1 2 u === 2 εεε0 E Potential difference and Electric field Since potential difference is work per unit charge, b ∆∆∆V === Edx ∫∫∫a For the parallel-plate capacitor E is uniform, so V === Ed Also for parallel-plate case Gauss’s Law gives Q Q εεε0 A E === σσσ /εεε0 === === Vd so C === === εεε0 A V d Spherical example A spherical capacitor has inner radius a = 3mm, outer radius b = 6mm. The charge on the inner sphere is q = 2 C. What is the potential difference? kq From Gauss’s Law or the Shell E === Theorem, the field inside is r 2 From definition of b kq 1 1 V === dr === kq −−− potential difference 2 ∫∫∫a r a b 1 1 1 1 === 9 ×××109 ××× 2 ×××10−−−9 −−− === 18 ×××103 −−− === 3 ×××103 V −−−3 −−−3 3 ×××10 6 ×××10 3 6 What is the capacitance? C === Q /V === 2( C) /(3000V ) === 7.6 ×××10−−−4 F A capacitor has capacitance C = 6 µF and charge Q = 2 nC. If the charge is Q.25-1 increased to 4 nC what will be the new capacitance? (1) 3 µF (2) 6 µF (3) 12 µF (4) 24 µF Q. -
Chapter 5 Capacitance and Dielectrics
Chapter 5 Capacitance and Dielectrics 5.1 Introduction...........................................................................................................5-3 5.2 Calculation of Capacitance ...................................................................................5-4 Example 5.1: Parallel-Plate Capacitor ....................................................................5-4 Interactive Simulation 5.1: Parallel-Plate Capacitor ...........................................5-6 Example 5.2: Cylindrical Capacitor........................................................................5-6 Example 5.3: Spherical Capacitor...........................................................................5-8 5.3 Capacitors in Electric Circuits ..............................................................................5-9 5.3.1 Parallel Connection......................................................................................5-10 5.3.2 Series Connection ........................................................................................5-11 Example 5.4: Equivalent Capacitance ..................................................................5-12 5.4 Storing Energy in a Capacitor.............................................................................5-13 5.4.1 Energy Density of the Electric Field............................................................5-14 Interactive Simulation 5.2: Charge Placed between Capacitor Plates..............5-14 Example 5.5: Electric Energy Density of Dry Air................................................5-15 -
Electric Potential
Electric Potential • Electric Potential energy: b U F dl elec elec a • Electric Potential: b V E dl a Field is the (negative of) the Gradient of Potential dU dV F E x dx x dx dU dV F UF E VE y dy y dy dU dV F E z dz z dz In what direction can you move relative to an electric field so that the electric potential does not change? 1)parallel to the electric field 2)perpendicular to the electric field 3)Some other direction. 4)The answer depends on the symmetry of the situation. Electric field of single point charge kq E = rˆ r2 Electric potential of single point charge b V E dl a kq Er ˆ r 2 b kq V rˆ dl 2 a r Electric potential of single point charge b V E dl a kq Er ˆ r 2 b kq V rˆ dl 2 a r kq kq VVV ba rrba kq V const. r 0 by convention Potential for Multiple Charges EEEE1 2 3 b V E dl a b b b E dl E dl E dl 1 2 3 a a a VVVV 1 2 3 Charges Q and q (Q ≠ q), separated by a distance d, produce a potential VP = 0 at point P. This means that 1) no force is acting on a test charge placed at point P. 2) Q and q must have the same sign. 3) the electric field must be zero at point P. 4) the net work in bringing Q to distance d from q is zero. -
A Vacuum Electrostatic Generator
1465 A VACUUM ELECTROSTATIC GENERATOR B.H. Choi, H.D. Kang, W. Kim, B.H. Oh, Korea Advanced Energy Research Institute Daeduk-Danji, Choongnam, 301-353, Korea and K.H. Chung Seoul National University Shinrim-Dong, Kwanak-Ku, Seoul, 151-741, Korea Abstract insulator supports the breakdown strength is limited to about 30 kV/cm due to the flashover phenomena on A compact electrostatic generator designed with the insulator surface. the principle of vacuum insulation has been developed. The concept of vacuum insulation instead of gas It consists of a rotating insulation disk with charge- insulation for the electrostatic generator design has carrying conductors placed around the circumference some advantages, such as the capability to hold the n n ti a non-contact induction system with R” electron high voltage in narrow inductor gap, elimination cf gun. The usable voltage of 130 kV with the generating the electrical cant acr problem by utilizing the current of about 300 pA has been obtained at the oper- non-contact induction method, posstbility of fice ational pressure of 1x10-’ torr. regulation of high voltage, and reduction cf frictional wind loss and the mechanical vibration. In Introduction addition, elimination of gas handling system may enhance the compactness and the flexibilities for the The recent progress of research and industrial accelerator system. The characteristics and the applications of ion beams require very stable beams operational performance of this vacuum electrostatic with the high energy of around 1 MeV and the current generator have been described in this paper. of a few mA. A candidate of high voltage sources to produce Experimental Apparatus mono-energetic beams is electrostatic generator, which has superior features such as small voltage ripple and The schematic drawins of the vacuum electrostatic small stored energy in the state of ultra high generator is shown in Fig. -
Basic Electrostatics System Model No
Instruction Manual Manual No. 012-07227E *012-07227* Basic Electrostatics System Model No. ES-9080 Basic Electrostatics System Model No. ES-9080 Table of Contents Equipment List........................................................... 3 Introduction .......................................................... 4-5 Equipment Description .............................................. 5-11 Electrometer...................................................................................................................................5 Electrostatics Voltage Source ........................................................................................................6 Variable Capacitor .........................................................................................................................7 Charge Producers and Proof Plane............................................................................................. 7-8 Proof Plane................................................................................................................................. 8-9 Faraday Ice Pail............................................................................................................................10 Conductive Spheres......................................................................................................................11 Resistor-Capacitor Network Accessory.......................................................................................11 Electrometer Operation and Setup Requirements................12-13 -
Physics 2102 Lecture 2
Physics 2102 Jonathan Dowling PPhhyyssicicss 22110022 LLeeccttuurree 22 Charles-Augustin de Coulomb EElleeccttrriicc FFiieellddss (1736-1806) January 17, 07 Version: 1/17/07 WWhhaatt aarree wwee ggooiinngg ttoo lleeaarrnn?? AA rrooaadd mmaapp • Electric charge Electric force on other electric charges Electric field, and electric potential • Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors • Electric currents Magnetic field Magnetic force on moving charges • Time-varying magnetic field Electric Field • More circuit components: inductors. • Electromagnetic waves light waves • Geometrical Optics (light rays). • Physical optics (light waves) CoulombCoulomb’’ss lawlaw +q1 F12 F21 !q2 r12 For charges in a k | q || q | VACUUM | F | 1 2 12 = 2 2 N m r k = 8.99 !109 12 C 2 Often, we write k as: 2 1 !12 C k = with #0 = 8.85"10 2 4$#0 N m EEleleccttrricic FFieieldldss • Electric field E at some point in space is defined as the force experienced by an imaginary point charge of +1 C, divided by Electric field of a point charge 1 C. • Note that E is a VECTOR. +1 C • Since E is the force per unit q charge, it is measured in units of E N/C. • We measure the electric field R using very small “test charges”, and dividing the measured force k | q | by the magnitude of the charge. | E |= R2 SSuuppeerrppoossititioionn • Question: How do we figure out the field due to several point charges? • Answer: consider one charge at a time, calculate the field (a vector!) produced by each charge, and then add all the vectors! (“superposition”) • Useful to look out for SYMMETRY to simplify calculations! Example Total electric field +q -2q • 4 charges are placed at the corners of a square as shown. -
Frequency Dependence of the Permittivity
Frequency dependence of the permittivity February 7, 2016 In materials, the dielectric “constant” and permeability are actually frequency dependent. This does not affect our results for single frequency modes, but when we have a superposition of frequencies it leads to dispersion. We begin with a simple model for this behavior. The variation of the permeability is often quite weak, and we may take µ = µ0. 1 Frequency dependence of the permittivity 1.1 Permittivity produced by a static field The electrostatic treatment of the dielectric constant begins with the electric dipole moment produced by 2 an electron in a static electric field E. The electron experiences a linear restoring force, F = −m!0x, 2 eE = m!0x where !0 characterizes the strength of the atom’s restoring potential. The resulting displacement of charge, eE x = 2 , produces a molecular polarization m!0 pmol = ex eE = 2 m!0 Then, if there are N molecules per unit volume with Z electrons per molecule, the dipole moment per unit volume is P = NZpmol ≡ 0χeE so that NZe2 0χe = 2 m!0 Next, using D = 0E + P E = 0E + 0χeE the relative dielectric constant is = = 1 + χe 0 NZe2 = 1 + 2 m!00 This result changes when there is time dependence to the electric field, with the dielectric constant showing frequency dependence. 1 1.2 Permittivity in the presence of an oscillating electric field Suppose the material is sufficiently diffuse that the applied electric field is about equal to the electric field at each atom, and that the response of the atomic electrons may be modeled as harmonic. -
Magnetic Fields and Forces
Magnetic Fields and Forces (1) Magnetic Fields are produced by moving charges. (2) Magnetic Fields exert forces on moving charges. (3) Stationary Charges do not produce magnetic fields. (4) Stationary Charges are not affected by stationary magnetic fields. Permanent Magnets There are naturally occurring magnetic substances. These were long used for navigation. As a result, we speak of “north” and “south” magnetic poles, instead of positive and negative magnetic charges. Permanent magnets can also be made by people from raw materials. These magnetic poles also respect an “opposites attract” rule: N S N S attraction N S S N repulsion Earth: North and South Revisited The ancients defined the “north” end of a permanent magnet as the end that is attracted to the “north” pole of the earth. points north S N S Therefore the north geographic pole of earth is a south magnetic pole. N Magnetic Field of a Bar Magnet This is a magnetic dipole field, very similar to the electric dipole field of two opposite electric charges. Note that magnetic field lines are all closed loops, unlike electric field lines that must start and stop on a charge. Important: There is no magnetic “monopole.” N Such a thing has never been observed. North and South magnetic poles always come in pairs of equal strength. Properties of the magnetic force on a charge moving in a B field 1. The magnetic force is proportional to the charge q and speed v of the particle 2. The magnitude and direction of the magnetic force depend on the velocity of the particle and on the magnitude and direction of the magnetic field. -
The Dipole Moment.Doc 1/6
10/26/2004 The Dipole Moment.doc 1/6 The Dipole Moment Note that the dipole solutions: Qd cosθ V ()r = 2 4πε0 r and Qd 1 =+⎡ θθˆˆ⎤ E ()r2cossin3 ⎣ aar θ ⎦ 4πε0 r provide the fields produced by an electric dipole that is: 1. Centered at the origin. 2. Aligned with the z-axis. Q: Well isn’t that just grand. I suppose these equations are thus completely useless if the dipole is not centered at the origin and/or is not aligned with the z-axis !*!@! A: That is indeed correct! The expressions above are only valid for a dipole centered at the origin and aligned with the z- axis. Jim Stiles The Univ. of Kansas Dept. of EECS 10/26/2004 The Dipole Moment.doc 2/6 To determine the fields produced by a more general case (i.e., arbitrary location and alignment), we first need to define a new quantity p, called the dipole moment: pd= Q Note the dipole moment is a vector quantity, as the d is a vector quantity. Q: But what the heck is vector d ?? A: Vector d is a directed distance that extends from the location of the negative charge, to the location of the positive charge. This directed distance vector d thus describes the distance between the dipole charges (vector magnitude), as well as the orientation of the charges (vector direction). Therefore dd= aˆ , where: Q d ddistance= d between charges d and − Q ˆ ad = the orientation of the dipole Jim Stiles The Univ. of Kansas Dept. of EECS 10/26/2004 The Dipole Moment.doc 3/6 Note if the dipole is aligned with the z-axis, we find that d = daˆz . -
01. Franklin Intro 9/04
Franklin and Electrostatics- Ben Franklin as my Lab Partner A Workshop on Franklin’s Experiments in Electrostatics Developed at the Wright Center for Innovative Science Teaching Tufts University Medford MA 02155 by Robert A. Morse, Ph.D. ©2004 Sept 2004 Benjamin Franklin observing his lightning alarm. Described in Section VII. Engraving after the painting by Mason Chamberlin, R. A. Reproduced from Bigelow, 1904 Vol. VII Franklin and Electrostatics version 1.3 ©2004 Robert A. Morse Wright Center for Science Teaching, Tufts University Section I- page 1 Copyright and reproduction Copyright 2004 by Robert A. Morse, Wright Center for Science Education, Tufts University, Medford, MA. Quotes from Franklin and others are in the public domain, as are images labeled public domain. These materials may be reproduced freely for educational and individual use and extracts may be used with acknowledgement and a copy of this notice.These materials may not be reproduced for commercial use or otherwise sold without permission from the copyright holder. The materials are available on the Wright Center website at www.tufts.edu/as/wright_center/ Acknowledgements Rodney LaBrecque, then at Milton Academy, wrote a set of laboratory activities on Benjamin Franklin’s experiments, which was published as an appendix to my 1992 book, Teaching about Electrostatics, and I thank him for directing my attention to Franklin’s writing and the possibility of using his experiments in teaching. I would like to thank the Fondation H. Dudley Wright and the Wright Center for Innovative Science Teaching at Tufts University for the fellowship support and facilities that made this work possible. -
Electromagnetic, Electrostatic and Magnetostatic Fields Electromagnetic Fields Are Characterized by Coupled, Dynamic (Time- Vary
Electromagnetic, Electrostatic and Magnetostatic Fields Electromagnetic fields are characterized by coupled, dynamic (time- varying) electric and magnetic fields and are governed by the complete set of Maxwell’s equations (four coupled equations). According to Maxwell’s equations, a time-varying electric field cannot exist without the a simultaneous magnetic field, and vice versa. Under static conditions, the time-derivatives in Maxwell’s equations go to zero, and the set of four coupled equations reduce to two uncoupled pairs of equations. One pair of equations governs electrostatic fields while the other set governs magnetostatic fields. This decoupling of Maxwell’s equations illustrates that static electric fields can exist in the absence of magnetic fields and vice versa. Stationary charges produce electrostatic fields while magnetostatic fields are produced by steady (DC) currents or permanent magnets. Maxwell’s Equations (electromagnetic fields) b ` Maxwell’s Equations Maxwell’s Equations (electrostatic fields) (magnetostatic fields) Electrostatic Fields Electrostatic fields are static (time-invariant) electric fields produced by static (stationary) charges. The mathematical definition of the electrostatic field is derived from Coulomb’s law which defines the vector force between two point charges. Coulomb’s Law Given point charges [q1, q2 (units=C)] in air located by vectors R1 and R2, respectively, the vector force acting on charge q2 due to q1 [F12 (units=N)] is defined by Coulomb’s law as where is a unit vector pointing from q1 to q2 and åo is the free-space !12 permittivity [åo = 8.854×10 F/m]. The permittivity of air is approximately equal to that of free space (vacuum).