Phys104

Chapter24Gauss’sLaw

Find the E-field a distance r from a line of positive charge of infinite length and constant charge per unit length λ.

Example24.7ACylindricallySymmetricChargeDistribu@on

•  The symmetry of the charge distribution requires that E be perpendicular to the line charge and directed outward.

•  SelectacylinderasGaussiansurface.Thecylinderhasaradiusofrandalengthofℓ.

•  E is constant in magnitude and perpendicular to the surface at each point.

•  The flux through the ends of the Gaussian cylinder is zero.

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Example24.7ACylindricallySymmetricChargeDistribu@on

Example24.8APlaneofChargeFindtheelectricfieldduetoaninfiniteplaneofposiEvechargewithuniformsurfacechargedensityσ.

•  By symmetry, E must be perpendicular to the plane and must have the same magnitude at all points equidistant from the plane.

•  A Gaussian surface that reflects the symmetry is a small cylinder whose axis is perpendicular to the plane and whose ends each have an area A.

•  The direction of E is away from positive charges indicates that the direction of E on one side of the plane must be opposite its direction on the other side.

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Example24.8APlaneofCharge•  BecauseEisperpendiculartodAeverywhereonthesurface,

andthereisnocontribuEontothesurfaceintegralfromthissurface.

•  ThefluxthrougheachendofthecylinderisEA;hence,thetotalfluxthroughtheenEreGaussiansurfaceisjustthatthroughtheends,

ΦE=2EA

24.4 Conductors in Electrostatic Equilibrium

" The electric field is zero everywhere inside the conductor. " Any net charge resides on the conductor’s surface. " The electric field just outside a charged conductor is

perpendicular to the conductor’s surface. " On an irregularly shaped conductor, the surface charge

density is greatest at locations where the radius of curvature of the surface is smallest.

When there is no net motion of charge within a conductor, the conductor is in electrostatic equilibrium. A conductor in electrostatic equilibrium has the following properties:

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24.4 Conductors in Electrostatic Equilibrium

"  The electric field is zero everywhere inside the conductor

Why is this so? •  Consider a conducting slab placed in an

external field E. •  If the field were not zero, free electrons in

the conductor would experience an electric force (F = q E) and would accelerate due to this force.

�  The charges in the conductor move creating an internal electric field that cancels the applied field on the inside of the conductor.

When electric charges are at rest, the electric field within a conductor is zero.

The electric field is stronger where the surface is more sharply curved.

On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest.

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• ChargeResidesontheSurface•  ChooseaGaussiansurfaceinsidebutcloseto

theactualsurface.•  Theelectricfieldinsideiszero.•  ThereisnonetfluxthroughtheGaussian

surface.•  BecausetheGaussiansurfacecanbeascloseto

theactualsurfaceasdesired,therecanbenochargeinsidethesurface.

Since no net charge can be inside the surface, any net charge must reside on the surface. Gauss’s law does not indicate the distribution of these charges, only that it must be on the surface of the conductor.

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E-Field’sMagnitudeandDirec@on

•  ChooseacylinderastheGaussiansurface.

•  Thefieldmustbeperpendiculartothesurface.–  IftherewereaparallelcomponenttoE,chargeswouldexperienceaforceandacceleratealongthesurfaceanditwouldnotbeinequilibrium.

�  Thenetfluxthroughthesurfaceisthroughonlytheflatfaceoutsidetheconductor.¡  Thefieldhereisperpendiculartothesurface

�  ApplyingGauss’slaw

E = σε00ε

σAEAE ==Φ

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