Presentation on theme: "Applications of Gauss’s Law"— Presentation transcript:
1 Applications of Gauss’s Law Lecture 9Applications ofGauss’s LawConductor inelectric field
2 ACT: Crossed planes A B C Which diagram corresponds to the E-field lines for these two uniformly charged infinite sheets that intersect each other as shown?+y+x+y+x+y+xA B C
3 Each sheet produces a uniform electric field. +y+y+x+xThe total E field is uniform in each quadrant.
4 EXAMPLE: Infinite line of charge A cable of diameter D = 3 mm and length L = 200 m has a total charge Q = 4 C uniformly distributed along its length. Find the electric field at point P, located at a distance r = 2 cm from it.PrL >> r → infinite cabler >> D → one-dimensional charge distribution
5 The elegant way (Gauss’s law) The system has cylindrical symmetry.The Gaussian surface should be a cylinder of radius r and height h.rPhLinear charge density:
8 Example: Line and sheet Find the magnitude of the electric field half-way between an infinite line of uniform charge (λ = +10 μC/m) that runs parallel to an infinite sheet of uniform charge (σ = +10 μC/m2). The distance between the line and the sheet is d = 1.0 m.λ = +10 μC/mσ = +10 μC/m2Pd/22.05 × 105 N/C4.10 × 105 N/C6.15 × 105 N/C9.25 × 105 N/C
9 Electric field produced by an infinite sheet at distance r : σ
10 Electric field produced by an infinite line at distance r: λ
11 σ = +10 μC/m2EsheetElinePλ = +10 μC/md/2d/2(Answer B)
12 The charge in a conductor in equilibrium is always on the surface(s). We know that E = 0 inside a conductor in equilibrium. Therefore, the electric flux though any Gaussian surface inside the conductor is zero.The charge enclosed by these surfaces is zero.There is no charge inside the conductor.The charge in a conductor in equilibrium is always on the surface(s).
13 Second application of Gauss’s Law: Finding the charge distribution when E is known.
14 Example: Conducting shell A solid non-conducting sphere with a total charge Q = +3 μC is surrounded by a concentric uncharged conducting spherical shell.What is the surface charge density σin on the inner surface of the shell?σoutσinQ
15 Negative charges are attracted towards the inner surface and positive charges are repelled towards the outer surface.σoutσinQ
16 Draw a Gaussian surface inside the conducting shell. σoutσinQRin
17 Charge on the outer surface: Q-QThe charge enclosed by the pink surface must be zero.
18 If the metal shell had a total charge of 3Q and the central charge was 100Q : The charge enclosed by the pink surface must be zero.If the metal shell had a total charge of 3Q :4Q-QQ
19 ACT: Conducting shell C. Ebefore > Eafter A. Ebefore < Eafter We now remove the uncharged shell. Compare the magnitude of the electric field at point P before and after the shell is removed.A. Ebefore < EafterB. Ebefore = EafterC. Ebefore > EafterσinQσoutσinQP
20 In both cases, the symmetry is the same In both cases, the symmetry is the same. We will use the same Gaussian surface: A sphere that contains point P.So the integral will look just the same:And the enclosed charge is also the same: Q.Therefore,σoutσinQP
21 ACT: Asymmetric conducting shell This time keep the shell, but move the internal charge off center. Compare the electric field at point P when the charge is centered/off center.QPA. Ecentered < Eoff-centerB. Ecentered = Eoff-centerC. Ecentered > Eoff-centerQP
22 Q -Q Q Q Flux through pink surface = 0 Total charge on inner surface is always -Q.QTotal charge on outer surface is always Q.Charge distribution on inner surface:UniformConcentrated near the sphereQQBut E = 0 inside the shell in both cases!Outer surface charge is uniform (Why wouldn’t it?)E is the same for both (that of a sphere with uniform charge Q)
24 ACT: Parallel charged planes II An uncharged metal slab is inserted between the two planes as shown below. Charge densities σL and σR appear on the sides of the slab. Compare the electric field at point P before and after the slab is introduced.σL + σR = 0and distance fromthe plane does not matter!Pσ1 = +4 μC/m2σLσ2 = -2 μC/m2σRA. Ebefore < Eafter B. Ebefore = Eafter C. Ebefore > Eafter
25 How can we determine σL and σR? The slab surfaces are two new charged planes. There are 4 contributions to the field inside the slabσ1σLσRσ2PNote: σ2 < 0, so that contribution really points in the opposite direction, but it’s easier to leave signs for the end.
26 • The net field inside the conducting slab is zero: σ1σLσRσ2P