Gauss’s Law Gauss’s law uses symmetry to simplify electric field calculations. Gauss’s law also gives us insight into how electric charge distributes itself over conducting bodies. Given any general distribution of charge, surround it with an imaginary surface that encloses the charge. Then examine the electric field at points on the imaginary surface. Gauss’s law is a relationship between all the points on the surface and the total charge enclosed by the surface.

Charge and Electric Flux Electric flux E is a measure of the number of electric field lines passing through a surface. When the surface being penetrated encloses some net charge, the net number of lines that go through the surface is proportional to the net charge within the surface.

Charge and Electric Flux

Charge and Electric Flux The previous slide shows an electric field that is uniform in both magnitude and direction. The electric field lines penetrate a rectangular surface of area A which is perpendicular to the field. The electric flux through this area is E = E•A Unit for E is

Charge and Electric Flux If the surface area A is not perpendicular to the field, the number of electric field lines passing through it must be less than that given by E = E•A When the normal (n hat ^ - I will be using A) to the surface of area A is at an angle θ to the uniform electric field, the number of lines that cross this area (A2) is equal to the number of lines that cross the area A1.

Charge and Electric Flux

Charge and Electric Flux The area vector A is perpendicular to the surface and always points outward.

Charge and Electric Flux Resolving the area vector A into an x and y component, the x- component is parallel to the electric field lines and the y-component is perpendicular to the electric field lines.

Charge and Electric Flux E = E·A·cos θ

Charge and Electric Flux The maximum flux E through a surface occurs when the surface is perpendicular to the electric field (or when the area vector A is parallel to the field, θ = 0).

Charge and Electric Flux The flux through a surface is zero when the surface is parallel to the field (or when the area vector A is perpendicular to the field, θ = 90.

Charge and Electric Flux When the electric field varies over the surface in question, the equation E = E·A·cos θ works only for a small area. Divide the area A into many small elements dA, determine the electric flux through each one, and integrate over the surface to obtain the total flux: E = E·dA

Charge and Electric Flux Electric flux is generally evaluated through a closed surface. A closed surface is a surface which divides space into an inside and an outside region, so that you cannot move from one region to the other without crossing the surface.

Charge and Electric Flux The surface of the sphere in this diagram is an example of a closed surface. The closed surface is often called a Gaussian surface.

Charge and Electric Flux The flux through a surface is positive if the angle between the area vector A and the electric field lines E is 0°; cos 0 = 1.

Charge and Electric Flux The flux through a surface is negative if the angle between the area vector A and the electric field lines E is 180°; cos 180 = -1.

Gauss’s Law Consider a positive point charge Q located at the center of a sphere of radius R as shown. The electric field E everywhere on the surface of the sphere is given by the En equation.

Gauss’s Law The electric field lines point radially outward from the surface and are perpendicular to the surface at each point, meaning that E is parallel to dA at each point ( = 0°). The area of a sphere is 4··r2 The electric flux through the surface is:

Gauss’s Law The net flux through a spherical Gaussian surface is proportional to the charge enclosed by the surface and does not depend on the radius of the sphere. The net flux through any closed surface is independent of the shape of the surface and is given by: E = 4·k·Qin

Gauss’s Law For a point charge located outside a closed surface, some electric field lines enter the surface and others leave the surface. The net electric flux through a closed surface that surrounds no charge is zero.

Gauss’s Law For several charges enclosed by a Gaussian surface S, the net flux depends on the net charge enclosed within the surface. The net charge within the Gaussian surface S is: Qnet = q1 +q2 +q3 = 1 + -1 + -1 = -1

Gauss’s Law When using E = 4·k·Qin, the Qin represents the net charge inside the Gaussian surface, the electric field term E represents the total electric field, which includes contributions from charge both inside and outside the Gaussian surface. In the previous example, the total electric field E would also include q4 and q5.

Gauss’s Law Electric field lines extend outward from a Gaussian surface if the enclosed charge is positive.

Gauss’s Law The electric field lines point inward when the enclosed charge is negative.