Chapter 3 - Electric Flux Density, Gauss's Law and Divergence

# Chapter 3 - Electric Flux Density, Gauss's Law and Divergence

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1 Chapter 3. Chapter 3. Electric Flux Density, Gauss's Law and Divergence Electric Flux Density, Gauss's Law and Divergence The Electric Flux Density Faraday's experiment: • A pair of concentric metallic spheres was constructed, the outer one consisting of two hemispheres that could be firmly clamped together. Shells of insulating (dielectric) material were also prepared which would occupy the entire volume between the concentric spheres. • With the equipment dismantled, the inner sphere was given a known positive charge. • The hemispheres were then clamped together around the charged sphere with about 2 cm of dielectric material between them • The outer sphere was discharged by connecting it momentarily to ground. • The outer sphere was separated carefully, using tools made of insulating material in order not to disturb the induced charge on it, and the negative induced charge on each hemisphere was measured.

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2 Outer Conducting Shell Insulating Filling Inner Conductor Faraday Faraday s Experiment s Experiment
3 Faraday found out that the total charge on the outer sphere was equal in magnitude to the original charge placed on the inner sphere and was true regardless of the dielectric material placed between the spheres. “Something” has caused charge to be induced at the outer conductor. He called it as displacement, displacement flux or electric flux Quantitatively, Ψ = Q, that is, the electric flux is directly proportional to the amount charge causing the flux (in SI units, they are equal, or the constant of proportionality is 1). The Electric Flux The Electric Flux

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4 Electric Flux Electric Flux The electric flux through a surface is a measure of the amount of electric field passing through a surface. For a closed surface: Electric Flux = Charge Enclosed
5 The Electric Flux Density The Electric Flux Density Define: D – Electric Flux Density – electric flux per unit area – unit: coulombs per square meter, C/m 2 Notes: D is a vector field belonging to the "flux density" class of vector fields. • The direction of D at a point is the direction of flux lines at that point, and the magnitude is equal to the number of flux lines crossing the surface normal to the lines divided by the surface area.

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6 Electric Flux Density due to a Spherical Charge Electric Flux Density due to a Spherical Charge D | r = a = 2 4 a Q π a r D | r = b = 2 4 b Q π a r D = 2 4 r Q π a r a < r < b b +Q -Q a
7 If the inner sphere is shrunk to a point charge while retaining its charge Q, then: D = 2 4 r Q π a r Comparing the equation above with E due to a point charge, then: E = 2 0 4 r Q πε a r In free space: D = ε 0 E .

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8 R P dV D = π ρ vol V R dv 2 4 a R Note: R is the vector from dv to the point where D is determined.
9 Gauss Gauss s Law s Law Gauss's Law The electric flux passing through any closed surface is equal to the total charge enclosed by that surface. Consider a small area of a closed surface:

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Chapter 3 - Electric Flux Density, Gauss's Law and Divergence

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