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Gauss law More fundamental statement about electric fields than Coulomb’s law.More fundamental statement about electric fields than Coulomb’s law. Easily calculate electric field for highly symmetric continuous charge distributions.Easily calculate electric field for highly symmetric continuous charge distributions. Understand the properties of conductors in electrostatic equilibrium.Understand the properties of conductors in electrostatic equilibrium.

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Overview Charge symmetry determines electric field symmetry. Calculate electric flux (number of electric field lines) through a surface. Gaussian surfaces: Just a closed surface around a charge. Gauss law: The electric flux through a Gaussian surface is related to the charge inside the surface.

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Symmetries The electric field must match the symmetry of the charge distribution!

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The symmetry argument Electric field must match the symmetry of the charge distribution!

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Concept Quiz What best describes the electric field symmetry outside the charged conducting sphere? A)Radially inward B)Radially outward C)Zero field D)Concentric - - - - - - - - - -

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Concept of flux Important analogy: Air flowing through a loop. The amount of air through the loop depends a) on the size of the loop, b) on the angle of the loop with respect to the air flow. Rate of air flow through the loop:

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Area as a vector quantity Vector is uniquely defined by a) magnitude b) direction. So we define the magnitude as the size of the area (π r 2,a x b, etc). And we define a direction by a unit vector perpendicular to the area

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Electric flux The electric flux due to an electric field E through area A = the number of electric field lines through A

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Electric flux If the electric field is not uniform, we have to sum up many small fluxes all over the area: The meaning of the surface integral that it sums over many small fluxes. It is a mathematical tool.

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1)Electric field everywhere tangent to surface: 2) Electric field everywhere perpendicular to surface and same magnitude E at every point:

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Flux through closed surface Definition: The area vector points from inside to outside

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Concept Quiz What is the electric flux through the cylinder with height h and radius r? A) Zero B) E 2 r h C) E (2 r 2 +2r h ) D) Eh r 2 h h r

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What is the electric flux passing through the cylinder? A.Positive C. Zero A.Negative

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What goes in, must come out! What is the electric flux passing through the cylinder? ^ n ^ n ^ n ^ n

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Gaussian surfaces Gaussian surface is a fictitious closed surface around a charge.

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Gauss law from Coulomb’s law The flux through the surface of the sphere: Coulomb’s law: Therefore r Q + Gauss surface ^ n

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Does hold for any closed surface? Q + Since the number of field lines through any closed surface is the same, Is true for any closed surface!

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Q1Q1 Q2Q2 Since the flux is related to the number of field lines passing through a surface the total flux is the flux from each charge Gauss Law

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Electric field of a long straight wire with charge density ¸ ++++++++++++ L r Gauss surface

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Concept Quiz What Gaussian surface would be best to apply Gauss law to the plane charge sheet? a)Sphere? b)Box? c)Cylinder?

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1)Electric field everywhere tangent to surface: 2) Electric field everywhere perpendicular to surface and same magnitude E at every point: ^ n

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Concept Quiz What is the electric field symmetry in the region between the spheres? A) Radially outward B) Radially inward C) Zero D) Concentric. +q -q

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Concept Quiz What is the electric field symmetry in the region outside the red sphere? A) Radially outward B) Radially inward C) Zero D) Concentric. +q -q

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Find the charge on the surface of the cavity and on the outer surface

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Find the charge on the surface of the hole and on the outer surface

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