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Explore the concepts of electric fields, conductors, insulators, equipotential surfaces, and more with examples like charged spheres and conducting plates. Understand the implications of Gauss’s Law and equipotential surfaces in studying electrostatics.
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Hw: All Chapter 5 problems and exercises Electricity and Magnetism Physics 208 Dr. Tatiana Erukhimova Lectures 14, 15
Outline • Applications of Gauss’s Law - The single Fixed Charge • Field of a sphere of charge • Field of a spherical shell • A Line of Charge • Conductors and Insulators • The electric field of a conductor • The field in the cavity of a conductor; Faraday’s Cage
Solid conducting sphere with charge Q A E V r A r
Electric field of a ball of charge Q Electric field outside of a charged sphere is exactly the same as the electric field produced by a point charge, located at the center of the sphere, with charge equal to the total charge on the sphere.
Electric field of a spherical shell Q The field outside the shell is like that of a point charge, while the field everywhere inside the shell is zero.
A Charged, Thin Sheet of Insulating Material + + + + + + + + + + +
Conductors and insulators Charges reside at the surface of the conductor + + + + + + + + Conductor + + E=0 + + + + +
What have we learned about conductors? • There is no electric field inside a conductor • Net charge can only reside on the surface of a conductor • Any external electric field lines are perpendicular to the surface (there is no component of electric field that is tangent to the surface). • The electric potential within a conductor is constant
Two parallel conducting plates - + + + - + a - + + d (the total field at any point between the plates)
An Apparent Contradiction - + + + - + - + +
An Apparent Contradiction - + + + - + - + + Near the surface of any conductor in electrostatics
1) There is a conducting spherical shell, inner radius A and outer radius B. If you put a charge Q on it, find the charge density everywhere. 2) There is a conducting spherical shell, inner radius A and outer radius B. A charge Q is put at the center. If you put a charge Q2 on the shell, find the charge density everywhere.
since inside the conductor. For any two points and inside the conductor The conductor’s surface is an equipotential.
Equipotential Surfaces An equipotential surface is a surface on which the electric potential V is the same at every point. Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge. So, electric field must be perpendicular to the surface at every point so that the electric force is always perpendicular to the displacement of a charge moving on the surface. Field lines and equipotential surfaces are always mutually perpendicular.
- - - - - - - - - - - Method of images: What is a force on the point charge near a conducting plate? Equipotential surface - -
The force acting on the positive charge is exactly the same as it would be with the negative image charge instead of the plate. The point charge feels a force towards the plate with a magnitude:
Equipotential surface - - - - - - - - - - - Method of images: A point charge near a conducting plane. - -
Equilibrium in electrostatic field: Earnshaw’s theorem There are NO points of stable equilibrium in any electrostatic field! How to prove it? Gauss’s Law will help! Imaginary surface surrounding P P If the equilibrium is to be a stable one, we require that if we move the charge away from P in any direction, there should be a restoring force directed opposite to the displacement. The electric field at all nearby points must be pointing inward – toward the point P. But that is in violation of Gauss’ law if there is no charge at P.
Thomson’s atom 1899 If charges cannot be held stably, there cannot be matter made up of static point charges (electrons and protons) governed only by the laws of electrostatics. Such a static configuration would collapse!
Capacitors Consider two large metal plates which are parallel to each other and separated by a distance small compared with their width. Area A L The field between plates is
Hw quiz An infinitesimally thin, insulating, uniformly charged horizontal sheet has a small charged object “floating” above it. If the object has mass m and charge Q, find σ, the charge per unit area on the sheet. Assume the sheet is of infinite extent.
