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Introduction of Gauss Law

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Gauss’s Law

What does it mean?

How do we use it?

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What does it mean?

  • First we need the concept of flux.

Area A

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What does it mean?

  • First we need the concept of flux.

Area A

Electric field

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What does it mean?

  • First we need the concept of flux.

Area A

Electric field

Flux is just electric field times area

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What does it mean?

  • First we need the concept of flux.

If electric field does not pass through the area, flux is zero.

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What does it mean?

  • First we need the concept of flux.

In general we use a normal vector to the plane, .

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What does it mean?

  • First we need the concept of flux.

For more general angles the flux varies as cosθ.

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What does it mean?

  • First we need the concept of flux.

For more general angles the flux varies as cosθ.

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What does it mean?

  • The total flux through a closed surface.

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What does it mean?

  • The total flux through a closed surface.
  • We use the convention that the normal always points outward.

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What does it mean?

  • The total flux through a closed surface.
  • We use the convention that the normal always points outward.

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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,

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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,

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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,

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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

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What does it mean?

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

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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.

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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.

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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 .

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What does it mean?

  • The full Gauss’s law.
  • The left side is the total flux out through the surface.

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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

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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, ε0, is the usual vacuum permittivity.

+q

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How do we use it?

  • For example, consider a charge +q.

+q

r

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How do we use it?

  • For example, consider a charge +q.
  • We choose a spherical surface.

+q

r

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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

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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

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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

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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

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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

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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

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How do we use it?

  • 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

+q

r

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How do we use it?

  • 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

+q

r

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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
  • constant and parallel to the normal, or
  • perpendicular to the normal.

​

  • Evaluate the total flux using the electric field as a free parameter.

​

r

+q

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Thanks