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Physics 272 January 27 Spring 2015 www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm go.hawaii.edu/KO Prof. Philip von Doetinchem [email protected] PHYS272 - Spring 15 - von Doetinchem - 128 Electric dipoles ● ● ● Pair of point charges with equal magnitude and opposite sign Important from molecules to antennas Example water: chemical bounds cause displacement of charge → water is a good solvent (e.g., salt splits to Na+ and Cl- in water) → very important for biochemical reactions PHYS272 - Spring 15 - von Doetinchem - 129 Force and torque of an electric dipole ● Electric dipole in a uniform external field Electric dipole moment points from the negative to the positive charge PHYS272 - Spring 15 - von Doetinchem - 130 Potential energy of an electric dipole (see chapter 10.4) =0: pot. energy minimal, stable equilibrium, dipole parallel to field =/2: pot. energy 0, dipole perpendicular to field = pot. energy maximum, dipole antiparallel to field ● Electric-field torque does work on dipole → change in potential energy ● Dipoles try to minimize potential energy ● An uncharged object with a dipole moment can experience a net force in a non-uniform electric field (polarization can happen due to electric field) PHYS272 - Spring 15 - von Doetinchem - 131 Electric flux and enclosed charge ● ● ● All electric field lines that enter an enclosed surface have to come out of the surface if there is no enclosed charge The number of electric field lines going through an area multiplied with the area is called flux Charges outside the enclosed surface do not give a net electric flux. PHYS272 - Spring 15 - von Doetinchem - 134 Flux of an uniform electric field PHYS272 - Spring 15 - von Doetinchem - 135 Flux of a nonuniform field ● ● General definition of electric flux: If flux is not uniform and area is curved → just integrate over infinitesimal area elements d General electric flux definition PHYS272 - Spring 15 - von Doetinchem - 136 Electric flux through a sphere ● ● Surface is not flat, electric field not uniform → make smart choice on enclosing surface to simplify the problem, make use of symmetries Electric field is perpendicular to surface PHYS272 - Spring 15 - von Doetinchem - 137 Electric flux through a sphere ● ● ● Radius cancels out → the flux through any surface enclosing a single point charge is independent of the shape or size of the surface Electric flux is independent of exact radius and only depends on enclosed charge. If you increase the size of the sphere, the electric field gets smaller, but the area increases → electric flux stays constant PHYS272 - Spring 15 - von Doetinchem - 138 Point charge inside a nonspherical surface ● ● Electric flux is positive (negative) where the electric field points out (into) of the surface Electric field lines can begin or end inside a region of space only when there is charge in that region PHYS272 - Spring 15 - von Doetinchem - 139 ● ● Total electric field is the vector sum of the electric fields of the individual charges inside a surface General form: Source: http://en.wikipedia.org/wiki/Carl_Gauss Gauß's law Carl Friedrich Gauß (1777 - 1855) ● The total electric flux through a closed surface is equal to the total (net) electric charge inside the surface, divided by 0. PHYS272 - Spring 15 - von Doetinchem - 140 Faraday's icepail experiment http://www.youtube.com/watch?v=GNizWxAD-9M http://youtu.be/GNizWxAD-9M?t=7m18s PHYS272 - Spring 15 - von Doetinchem - 141 Faraday's icepail experiment ● ● Rod inside pail: – Position of rod does not matter: reading of electroscope does not change – Charge on inside wall opposite of outside wall – Charge of pail is the same as the rod – Take rod out: electroscope reads zero remove electrons on surface (e.g., by touching) remove rod Conducting ball touches pail: – Charge transfers to surface of pail – Take ball out: no charge on ball, but charge on pail surface PHYS272 - Spring 15 - von Doetinchem - 142 Electric field of a conductor Charged conducting object ● ● We know that the electric field inside a conductor is zero (excess charges would move otherwise) Charges can only sit on the surface: – Construct Gaussian surface inside conducting object: electric field zero → no charge inside – You can do this for an infinite number of Gaussian surfaces inside the conductor → charge is always zero → Excess charges on a solid conductor resides entirely on the surface, and not in the interior. PHYS272 - Spring 15 - von Doetinchem - 145 Field of a charged conducting sphere surface integral apply Gauss' law ● Field inside is zero in a conductor ● Field outside is the same as for a point charge PHYS272 - Spring 15 - von Doetinchem - 146 Field of a charged insulating sphere apply Gauss' law surface integral ● Electric field is a continuous function of the radius (in contrast to conductor) PHYS272 - Spring 15 - von Doetinchem - 147 Van de Graaf generator http://www.youtube.com/watch?v=sy05B32XTYY PHYS272 - Spring 15 - von Doetinchem - 149 Van de Graaf generator ● Can be used to build up very high charges on top of shell surface PHYS272 - Spring 15 - von Doetinchem - 150 Faraday cage http://www.youtube.com/watch?v=WqvImbn9GG4 PHYS272 - Spring 15 - von Doetinchem - 151 Faraday cage ● ● Faraday cages protect against the external influence of electric fields All charges in a conductor reside on the outer surface, and always rearrange themselves to cancel out the electric field in the interior. Vacuum chamber: Electric noise from pumps → distorts measurement Sensitive silicon detector PHYS272 - Spring 15 - von Doetinchem - 152 Electric field inside a Hydrogen atom ● ● Hydrogen atom: proton surrounded by electron (treat both point-like) Electron moves → charge is smeared out → position probability follows an exponential function PHYS272 - Spring 15 - von Doetinchem - 153 ● ● ● Ratio of electric field of Hydrogen and the proton alone E/Eproton Electric field inside a Hydrogen atom Study the influence and relative field strength electron+proton field goes much quicker to zero than than proton alone r/a0 PHYS272 - Spring 15 - von Doetinchem - 154 Electric potential energy ● ● ● Charged particle moving in a field: field exerts work on particle Work can be expressed as potential energy: position of a charge in an electric field Use electric potential to describe potential electric energy → potential differences are important for understanding of electric circuits ● Work done on a particle to move from a to b: ● change of potential energy for a conservative force (reversible): PHYS272 - Spring 15 - von Doetinchem - 155 Electric potential energy in an uniform field a d b ● ● ● Conservative force → independent of exact path Potential energy decreases if a charged particle moves in the direction of the electric field If the displacement of a positive charge is in the direction of the electric field the work is positive PHYS272 - Spring 15 - von Doetinchem - 156 Electric potential energy of two point charges ● Potential energy of two point charges ● Potential energy defined to some reference point ● ● – This constant reference point should be smartly chosen depending on the problem – Very often only the difference of two potential energy levels matter, which makes the reference point irrelevant Potential energy is a shared property of both charges Electric field is the vector sum and the total potential energy is the algebraic sum (potential energy is NOT a vector): PHYS272 - Spring 15 - von Doetinchem - 157 Electric potential ● ● Describe potential energy on a “per unit charge” basis (like the electric field describes force per unit charge) Determination of electric field is often easier by using the potential Source: http://de.wikipedia.org/wiki/Alessandro_Volta Alessandro Volta 1745-1825 ● Potential energy and potential are scalars ● Potential difference in circuits is often called voltage PHYS272 - Spring 15 - von Doetinchem - 159 Finding electric potential from electric field ● If electric field is known or can be found easily: ● Potential does not depend on the exact path ● ● Moving with the direction of the electric field means moving in the direction of decreasing potential Moving a charge slowly against an electric field requires an external force, equal and opposite to the electric force. PHYS272 - Spring 15 - von Doetinchem - 160 Electron volts ● ● Electric field unit: 1V/m = 1 Volt/meter = 1N/C = 1 Newton/Coulomb Useful: electron charge → electron volt unit of energy ● For instance, very important in particle physics and to describe processes in atoms PHYS272 - Spring 15 - von Doetinchem - 161 Electric force and electric potential Proton moves in a uniform electric field: Proton feels electric force =1 (because proton moves in the direction ofPHYS272 the electric lines) - Springfield 15 - von Doetinchem - 162 Electric force and electric potential PHYS272 - Spring 15 - von Doetinchem - 163