Download Electric Flux

LECTURE 2
ELECTRIC FIELD / GAUSS’S LAW
Scope of the lecture:
1.
2.
3.
4.
What is meant by electric flux?
What is Gauss' law?
How can Gauss' law be used to calculate the electric field of a point charge?
What can you learn about the electric field inside an isolated conductor from Gauss'
law?
5. How to use Gauss' law to calculate the electric field near a charged conducting
surface?
6. How to use Gauss' law to calculate the electric field:
• near a non-conducting sheet of charge
• around an infinite line charge
• between oppositely charged conducting plates
• outside and inside a conducting sphere
• outside and inside a uniformly charged non-conducting sphere
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Electric Flux
The electric flux through a planar area (Fig.1a) is defined as the electric field times the
component of the area perpendicular to the field. If the area is not planar (Fig.1b), then the
evaluation of the flux generally requires an area integral since the angle will be continually
changing.
Note: When the area A is used in a vector operation like this, it is understood that the
magnitude of the vector is equal to the area and the direction of the vector is perpendicular to
the area.
Fig.1a
Physics for Engineers II / M. Mulak / WUT
Fig.1b
LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Area Integral (general remarks)
An area integral of a vector function E can be defined as
the integral on a surface of the scalar product of E with
area element dA. The direction of the area element is
defined to be perpendicular to the area at that point on
the surface.
The outward directed surface
integral over an entire closed
surface is denoted by
Gauss's Law
The total of the electric flux out of a closed surface is equal to the charge
enclosed divided by the permittivity.
Gauss's Law is a general law applying to any closed surface.
It is an important tool since it permits the assessment of the amount of enclosed charge by
mapping the field on a surface outside the charge distribution.
For geometries of sufficient symmetry it simplifies the calculation of the electric field.
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LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Gauss' Law, Integral Form
The area integral of the electric field over any closed
surface is equal to the net charge enclosed in the surface
divided by the permittivity of space.
Gauss' law is a form of one of Maxwell's equations, the
four fundamental equations for electricity and
magnetism.
Note: Gauss' law permits the evaluation of the electric field in many practical situations by
forming a symmetric Gaussian surface surrounding a charge distribution and evaluating
the electric flux through that surface.
Example: Electric Field of Point Charge
The electric field of a point charge Q can be obtained by a
straightforward application of Gauss' law. Considering a Gaussian
surface in the form of a sphere at radius r, the electric field has the
same magnitude at every point of the sphere and is directed outward.
The electric flux is then just the electric field times the area of the
sphere.
The electric field at radius r is then given by:
If another charge q is placed at r, it would experience a force
so this is seen to be consistent with Coulomb's law.
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LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Conductor at Equilibrium
For a conductor at equilibrium:
1. The net electric charge of a conductor resides entirely on its surface. (The mutual
repulsion of like charges from Coulomb's Law demands that the charges are as far apart as
possible, hence on the surface of the conductor.)
2. The electric field inside the conductor is zero. (Any net electric field in the conductor
would cause charge to move since it is abundant and mobile. This violates the condition of
equilibrium: net force =0.)
3. The external electric field at the surface of the conductor is perpendicular to that
surface. (If there were a field component parallel to the surface, it would cause mobile charge
to move along the surface, in violation of the assumption of equilibrium.)
The above facts are employed below:
Electric Field: Conductor Surface
Examining the nature of the electric field
near a conducting surface is an important
application of Gauss' law. Considering a
cylindrical Gaussian surface oriented
perpendicular to the surface, it can be seen
that the only contribution to the electric flux
is through the top of the Gaussian surface.
The flux is given by
The fact that the conductor is at equilibrium is
and the electric field is simply
an important constraint in this problem. It tells
us that the field is perpendicular to the surface,
because otherwise it would exert a force parallel
to the surface and produce charge motion.
Likewise it tells us that the field in the interior
of the conductor is zero, since otherwise charge While strictly true only for an infinite
would be moving and not at equilibrium.
conductor, it tells us the limiting value as
we approach any conductor at equilibrium.
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LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Applications of Gauss' Law
Gauss' law is a powerful tool for the calculation of electric fields when they originate from
charge distributions of sufficient symmetry to apply it. A few examples:
Electric Field: Sheet of Charge
For an infinite sheet of charge, the electric field will be perpendicular to the surface.
Therefore only the ends of a cylindrical Gaussian surface will contribute to the electric flux .
The resulting field is half that of a conductor at equilibrium with this surface charge density.
Physics for Engineers II / Maciej Mulak
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LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Electric field of line charge with a uniform linear charge density
Considering a Gaussian surface in the form of a cylinder at
radius r, the electric field has the same magnitude at every
point of the cylinder and is directed outward. The electric
flux is then just the electric field times the area of the
cylinder.
Electric Field: Parallel Plates
If oppositely charges parallel conducting plates are treated like infinite planes (neglecting
fringing), then Gauss' law can be used to calculate the electric field between the plates.
Presuming the plates to be at equilibrium with zero electric field inside the conductors, then
the result from a charged conducting surface can be used:
This is also consistent with treating the charge layers as two charge sheets
with electric field
in both directions.
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LECTURE 2 / ELECTRIC FIELD: GAUSS’S LAW
Electric Field of Conducting Sphere
Again, a straightforward application of Gauss' law: considering a
Gaussian surface in the form of a sphere at radius r > R , the
electric field has the same magnitude at every point of the surface
and is directed outward. The electric flux is then just the electric
field times the area of the spherical surface.
Note: The electric field is seen to be identical to that of a point charge Q
at the center of the sphere. Since all the charge will reside on the
conducting surface, a Gaussian surface at r< R will enclose no charge,
and by its symmetry can be seen to be zero at all points inside the
spherical conductor
Electric Field: Sphere of Uniform Charge (an insulator)
Now, considering a Gaussian surface in the form of a sphere at
radius r > R, the electric field has the same magnitude at every point
of the surface and is directed outward. The electric flux is then just
the electric field times the area of the spherical surface.
The electric field outside the sphere (r > R)is seen to be identical to that
of a point charge Q at the center of the sphere.
But: for a radius r < R, a Gaussian surface will enclose less than the
total charge and the electric field will be less. Inside the sphere of charge,
the field is given by:
Exercise: show that the last formula is true!
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