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CYK/2007/PH 410/Quiz 1
Electrodynamics I
- A sphere of radius
carries a charge density
(where
is a constant). Find the energy of the configuration.[4]
- A point charge
of mass
is released from rest at a distance
from an infinite grounded conducting plane. How long will it take for the charge to hit the plane?[3]
- Consider a charge free volume
bounded by a closed surface
that consists of several separate surfaces (Conductors)
each held at potential
. Let
be a well behaved function in
and on S, with a value equal to
on each surface
, but otherwise arbitraty for the present. Define the quantity
Prove using variational calculus:
, which is nonnegative by definition, is stationary and an absolute minimum if and only if
satisfies the Laplace equation in
and takes on the specified values
on the surfaces
.[3]
Answers
- Answer 1
- The electric field for the given charge density can be found by the Gauss law
|
(1) |
Hence the energy is given by
Alternatively
The Potential at any
is given by
|
(5) |
And then the energy is given by
- Answer 2
- From the method of images, the force on charge
is
|
(9) |
By Newton's Law,
Integrating further, we get,
|
(12) |
- Answer 3
- For a small change in function
, the change in the functional
is given by (to the first order)
The surface integral vanishes, since
on surface. Hence
will vanish for all
if and only if
.
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Charudatt Kadolkar
2007-02-11