Gauss’s Law Dominic Berry University of Waterloo Griffith University 8 February, 2011 What does it mean? How do we use it?

What does it mean? First we need the concept of flux. Area A

What does it mean? First we need the concept of flux. Area A Electric field

What does it mean? First we need the concept of flux. Area A Electric field Flux is just electric field times area

What does it mean? First we need the concept of flux. If electric field does not pass through the area, flux is zero.

What does it mean? First we need the concept of flux. In general we use a normal vector to the plane,.

What does it mean? First we need the concept of flux. For more general angles the flux varies as cos .

What does it mean? First we need the concept of flux. For more general angles the flux varies as cos .

What does it mean? The total flux through a closed surface.

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward.

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward.

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward. For the four sides,

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward. For the four sides, For the top,

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward. For the four sides, For the top, For the bottom,

What does it mean? The total flux through a closed surface. We use the convention that the normal always points outward. For the four sides, For the top, For the bottom, The total flux is

What does it mean? What does the integral mean? The circle indicates an integral over the closed surface.

What does it mean? What does the integral mean? The circle indicates an integral over the closed surface. In practice we will not have to evaluate the interval.

What does it mean? What does the integral mean? The circle indicates an integral over the closed surface. In practice we will not have to evaluate the interval. We break the surface up into sections where the flux is easy to calculate.

What does it mean? What does the integral mean? The circle indicates an integral over the closed surface. In practice we will not have to evaluate the interval. We break the surface up into sections where the flux is easy to calculate. In principle sum over infinitesimal elements.

What does it mean? The full Gauss’s law. The left side is the total flux out through the surface.

What does it mean? The full Gauss’s law. The left side is the total flux out through the surface. The right side is proportional to the charge, q, inside the surface. +q

What does it mean? The full Gauss’s law. The left side is the total flux out through the surface. The right side is proportional to the charge, q, inside the surface. The constant,  0, is the usual vacuum permittivity. +q

How do we use it? For example, consider a charge  q. +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. Gauss’s law gives +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. Gauss’s law gives +q r

How do we use it? For example, consider a charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. Gauss’s law gives +q r

How do we use it? +q Consider a shell of charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. Gauss’s law gives r

How do we use it? +q Consider a shell of charge  q. We choose a spherical surface. By spherical symmetry the electric field must be directed radially outwards. The magnitude of the electric field must be constant on the surface. The flux is just EA. Gauss’s law gives r

How do we use it? General procedure: Choose a surface corresponding to the symmetry of the problem. Break the surface up into subsurfaces where the electric field is either 1. constant and parallel to the normal, or 2. perpendicular to the normal. Evaluate the total flux using the electric field as a free parameter. Solve Gauss’s law for E. r +q