Gauss' Law
Although the electric field of any given charge distribution can be
calculated using Coulomb's Law (as we did last week), the method involves
the evaluation of integrals, which, to put it mildly, is tedious.
Happy Happy Joy Joy there exists another method of calculating the
electric field, which relies on a theorem called Gauss' Law. This law is a
consequence of Coulomb's Law and there is no new physics but it does
contain some new math which provides an elegant shortcut for calculating
the electric field of a charge distribution PROVIDED that the distribution
has a certain amount of symmetry. Gauss' Law doesn't work in all cases
but when it does it is wondrous.
However, we need first to introduce a new concept: Electric Flux
Electric Flux and the number of field lines
Consider a mathematical
(that is, imagined surface
in the shape of a rectangle
of area A that is immersed
in a constant electric field
E. This electric field
makes an angle
θ
with the
surface and therefore has
a component tangential to
the surface and a
component normal, that is
perpendicular, to the
surface. The
electric
flux
Φ
though the surface
is defined as the product of the area A by the magnitude of the normal
component of the electric field
Φ
= E
n
A = E A cos
θ
The quantity A cos
θ
can be seen as the projection of the area A onto a
plane perpendicular to the electric field, that is, A cos
θ
can be regarded
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View Full Documentas that part of the area A the faces the electric field 'head on.'
Now the magnitude of the electric field must be proportional to the
magnitude of the charge, the number of field lines that we draw coming
out of a (positive) charge must be proportional to the charge. There is a
convention that the number of field lines coming out of a charge Q is Q/
ε
o
 so the number of field lines coming out of a coulomb of charge is 1/
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 Spring '09
 G.R.GRIST
 Charge, Electrostatics, Gauss' Law, Electric charge

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