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Transcript
Lecture 3 β’ β’ Electric Flux Gaussβs Law Electric Flux We have used electric field lines to visualize electric fields and indicate their strength. Case one: The electric flux for a plan surface perpendicular to a uniform electric field We are now going to count the number of electric field lines passing through a surface, and use this count to determine the electric field. E Example What is electric flux F for closed cylinder of radius R immersed in a uniform electric field as shown in figure? Solution Since E is constant then F = - EA + 0 + EA = zero Electric Flux ΙΈπ¬ = π¬. π π¨ Example #2 β’ Consider a uniform electric field E oriented in the x direction. Find the net electric flux through the surface of a cube of edge length l, oriented as shown in the figure. The Electric Flux due to a point charge To calculate the electric flux due to a point charge we consider an imaginary closed spherical surface with the point charge in the center, this surface is called gaussian surface. Then the flux is given by (q = 0) ΙΈ= π¬. π π¨ = π¬ π ΙΈ= ππ πΊπ ππ π π¨ ππ¨π¬ π½ π π π π¨ = ππ π ππ πΊπ ππ π ΙΈ= πΊπ Note that the net flux through a spherical gaussian surface is proportional to the charge q inside the surface. Gaussian surface Consider several closed surfaces as shown in figure 1 , surrounding a charge Q as in the figure below. The flux that passes through surfaces S1, S2 and S3 all has a value q/ Ιo. Therefore we conclude that the net flux through any closed surface is independent of the shape of the surface Consider a point charge located outside a closed surface as shown in figure 2. We can see that the number of electric field lines entering the surface equal the number leaving the surface. Therefore the net electric flux in this case is zero, because the surface surrounds no electric charge. Example 4.2 In figure below, there are two equal and opposite charges of 2Q and -2Q what is the flux F for the surfaces S1, S2, S3 and S4. Solution For S1 the flux F = zero For S2 the flux F = zero For S3 the flux F = +2Q/ Ι0 For S4 the flux F = -2Q/ Ι0 Gaussβs Law Gauss law is a very powerful theorem, which relates any charge distribution to the resulting electric field at any point. As we saw the electric field lines means that each charge q must have q/eo flux lines coming from it. This is the basis for an important equation referred to as Gaussβs law. Note the following facts: 1. If there are charges q1, q2, q3, ......qn inside a closed (gaussian) surface, the total number of flux lines coming from these charges will be (q1 + q2 + q3 + ....... +qn)/Ι0 2. The number of flux lines coming out of a closed surface is the integral of over the surface, π¬. π π¨ So we can say: πππ π¬. π π¨ = πΊπ where qin is the total charge inside the gaussian surface. Gaussβs law states that the net electric flux through any closed gaussian surface is equal to the net electric charge inside the surface divided by the permittivity.