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Physics 272 September 9 Fall 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html Prof. Philip von Doetinchem [email protected] Phys272 - Fall 14 - von Doetinchem - 120 Summary ● Electric dipole in an external field Phys272 - Fall 14 - von Doetinchem - 121 Potential energy of an electric dipole =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 - Fall 14 - von Doetinchem - 122 Electric flux and enclosed charge ● ● If there is no enclosed charged: electric field flux going into the surface cancels flux out of the surface. Charges outside the enclosed surface do not give a net electric flux. Phys272 - Fall 14 - von Doetinchem - 125 Flux of an uniform electric field Phys272 - Fall 14 - von Doetinchem - 126 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 - Fall 14 - von Doetinchem - 127 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 Radius cancels out → the flux through any surface enclosing a single point charge is independent of the shape or size of the surface Phys272 - Fall 14 - von Doetinchem - 128 ● Gauß's law is an alternative to Coulomb's law and is a different way to express the relationship between electric charge and electric field. Source: http://en.wikipedia.org/wiki/Carl_Gauss Gauß's law Carl Friedrich Gauß (1777 - 1855) ● ● 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 - Fall 14 - von Doetinchem - 129 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 - Fall 14 - von Doetinchem - 130 General form of Gauß's law ● ● ● ● Surface encloses multiple charges Total electric field is the vector sum of the electric fields of the individual charges General form: The total electric flux through a closed surface is equal to the total (net) electric charge inside the surface, divided by 0. Phys272 - Fall 14 - von Doetinchem - 131 Faraday's icepail experiment http://www.youtube.com/watch?v=GNizWxAD-9M http://youtu.be/GNizWxAD-9M?t=7m18s Phys272 - Fall 14 - von Doetinchem - 132 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 - Fall 14 - von Doetinchem - 133 Applications of Gauß's Law ● ● ● ● If charge distribution is known and has enough symmetry the integral can be evaluated If we know the field we can find the charge distribution. Typical question: what is the electric field caused by a charge distribution on a conductor. Excess charge at rest on a solid conductor resides entirely on the surface, and not in the interior. Phys272 - Fall 14 - von Doetinchem - 136 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 - Fall 14 - von Doetinchem - 137 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 - Fall 14 - von Doetinchem - 138 Faraday cage http://www.youtube.com/watch?v=WqvImbn9GG4 Phys272 - Fall 14 - von Doetinchem - 139 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 - Fall 14 - von Doetinchem - 140 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 - Fall 14 - von Doetinchem - 142 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 - Fall 14 - von Doetinchem - 143 Electric potential energy of two point charges ● Potential energy of two point charges ● Potential energy defined to some reference point ● 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 - Fall 14 - von Doetinchem - 144 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 - Fall 14 - von Doetinchem - 146