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# chapter24 - Chapter 24 Gauss's Law Electric Flux Electric...

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Chapter 24 Gauss’s Law

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Electric Flux z Electric flux is the product of the magnitude of the electric field and the surface area, A , perpendicular to the field z Φ E = EA
Electric Flux, General Area z The electric flux is proportional to the number of electric field lines penetrating some surface z The field lines may make some angle θ with the perpendicular to the surface z Then Φ E = EA cos θ

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Electric Flux, Interpreting the Equation z The flux is a maximum when the surface is perpendicular to the field z The flux is zero when the surface is parallel to the field z If the field varies over the surface, Φ = EA cos θ is valid for only a small element of the area
Electric Flux, General z In the more general case, look at a small area element z In general, this becomes cos Ei i ii i EA θ ΔΦ = Δ = ⋅ Δ rr 0 surface lim i i A E d Δ→ Φ= Δ

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Electric Flux, final z The surface integral means the integral must be evaluated over the surface in question z In general, the value of the flux will depend both on the field pattern and on the surface z The units of electric flux will be N . m 2 /C 2
Electric Flux, Closed Surface z Assume a closed surface z The vectors point in different directions z At each point, they are perpendicular to the surface z By convention, they point outward i Δ A r

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Flux Through Closed Surface, cont. z At (1), the field lines are crossing the surface from the inside to the outside; θ < 90 o , Φ is positive z At (2), the field lines graze surface; θ = 90 o , Φ = 0 z At (3), the field lines are crossing the surface from the outside to the inside;180 o > θ > 90 o , Φ is negative
Flux Through Closed Surface, final z The net flux through the surface is proportional to the net number of lines leaving the surface z This net number of lines is the number of lines leaving the surface minus the number entering the surface z If E n is the component of E perpendicular to the surface, then En dE d A Φ= = ∫∫ EA rr Ñ Ñ

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Flux Through a Cube, Example z The field lines pass through two surfaces perpendicularly and are parallel to the other four surfaces z For side 1, E = -E l 2 z For side 2, E = E 2 z For the other sides, E = 0 z Therefore, E total = 0
Karl Friedrich Gauss z 1777 – 1855 z Made contributions in z Electromagnetism z Number theory z Statistics z Non-Euclidean geometry z Cometary orbital mechanics z A founder of the German Magnetic Union z Studies the Earth’s magnetic field

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Gauss’s Law, Introduction z Gauss’s law is an expression of the general relationship between the net electric flux through a closed surface and the charge enclosed by the surface z The closed surface is often called a gaussian surface z Gauss’s law is of fundamental importance in the study of electric fields
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chapter24 - Chapter 24 Gauss's Law Electric Flux Electric...

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