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## Gauss’ Law

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**Gauss’ Law**• Electric Field Lines / Electric Field Vectors • Electric Flux • Gauss’ Law • Use of Gauss’ Law and Gaussian Surfaces • Electrostatic Equilibrium • Conductors • Non Conductors**Electric Force and Acceleration**• The electric force is • given by • F = qE • The acceleration by q = a E m**Electric Flux**• A measure of the amount of electric field through an area perpendicular to the field • The “number” of field lines through the area.**Area Vector**Define Area Vector**Definition of symbols**A= Area (always positive number) n= Unit vector. Its direction corresponds to the orientation of the area Forms a right handed system**Dot product Definition of Flux**• Electric Flux • Number of Field lines • through Perpendicular surface**Flux through closed surface**• Flux through a • closed • surface from an • external source is zero**Flux through Curved Surface**ò F = · E A d surface ( ) · = q E d EdA A Cos ò = A dA surface**Gaussian Surface**• Gaussian Surface defined as • Surface • surrounding charge • where magnitude of Electric Field is constant or zero • the direction of Electric Field is same as the Area vectors of the surface • thus same symmetry as charge distribution**Flux through any closedsurface surrounding a charge is the**same**Gauss' Law I**( ) E r ò F = · E A d Gaussian surface ò ( ) = E r dA Gaussian surface ò = dA Gaussian surface ( ) r = p E r 4 2**Gauss' Law III**Using Coulombs Law for a point charge Q = p r k 4 2 r 2 Q = = p 4 kQ e 0**Gauss’ Law**Gauss' Law II ò F = · E A d Gaussian surface Q = e 0**Use of Gauss' Law**To Find Electric Field of Given Charge Distribution Surface + Charge Field**Coulombs Law from Gauss' Law I**Gauss' Law Coulombs' Law**Electrostatic Equilibrium**Electrostatic Equilibrium for objects in an external Electric Field • Conductors • No net motion of charge within conductor • Non Conductors • in non conductors there is no movement of charge • therefore always have equilibrium**At ElectrostaticEquilibrium**At Electrostatic Equilibrium • Electric Field is zero within conductor • Any excess charge on an isolated conductor must be on its surface • accumulates at points where radius of curvature is greatest**Electric Field just outsideconductor**• is perpendicular to conductors surface • has magnitude = • surface density / permitivity**Electric Field inside conductor**• Net Electric Field is zero inside, • otherwise Net Electric Force on charges • which then accelerate and move charges (on the average)**Why is the Charge on the Surface?**Why is the charge on the surface? Gaussian Surface 1 E=0 Q Gaussian Surface 2 Use Gauss’ Theorem**Answer**Charge must be between surface 1 and surface 2 (why?) Therefore must be on the surface of object**Answer**• Zero Flux through 2 • Zero Flux through 3 • Only Flux through 1 2 3 1 E**Answer 2**Q inside ò = · E A d cylinder e cylinder 0 ò ( ) = E r dA disk 1 ( ) = E r A Q ( ) s r inside ( ) \ = = E r cylinder e e A 0 0**Answer 3**Direction of Field? • Must be orthogonal to surface • otherwise there will be net motion on surface**Graph of Field v. Position**magnitude of electric field radius of conductor distance from center of charged conductor**Conductor in Electric Field**• In external field conductor • becomes polarized • InducedElectric Field from the surface must cancel external Electric Field inside conductor**Induced Field**E +dq E -dq Einduced -dq +dq E -dq +dq E**Charged Conductor**• If the conductor has a net charge • then it is also a source of an Electric Field • that combines with the external field • producing a resultant field • external to the conductor**Electric Field inside Cavities**Electric Fields inside Cavities of Conductors Gaussian Surface Cavity**Analysis 1**• Total charge within Gaussian surface must be zero • Otherwise there is an Electric Field inside the conductor around the cavity**Analysis 2**• Therefore NO charge on surface of cavity • Can enlarge cavity so that conductor is hollow • Faraday cage**Thought Question**Radio reception over some bridges**Electric Field inside Nonconductor**Electric Field inside non conductor?**Graph of Field v. Position**magnitude of electric field radius of non conductor distance from center of charged non conductor**Field Above Conductor**Field above surface of charged conductor Q s = = E e e A 0 0 Does not depend on thickness of conductor**Field Above Very Thin Nonconductor**Field above surface of charged nonconductor