Due: 11:55pm on Wednesday, April 6, 2011
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Are Coulomb Forces Conservative?
Learning Goal:
To review the concept of conservative forces and to understand that electrostatic forces are, in fact, conservative.
As you may recall from mechanics, some forces have a very special property, namely, that the work done on an object does not depend on the
object's trajectory; rather, it depends only on the initial and the final positions of the object.
Such forces are called
conservative forces
. If only conservative forces act within a closed system, the total amount of mechanical energy is
conserved within the system (hence the term "conservative"). Such forces have a number of properties that simplify the solution of many
problems.
You may also recall that a
potential energy function
can be defined with respect to a conservative force. This property of conservative forces will
be of particular interest of us.
Not all forces that we deal with are conservative, of course. For instance, the amount of work done by a frictional force very much depends on the
object's trajectory. Friction, therefore, is not a conservative force. In contrast, the gravitational force and the normal force are examples of
conservative forces. What about electrostatic (Coulomb) forces? Are they conservative, and is there a potential energy function associated with
them?
In this problem, you will be asked to use the given diagram to calculate the work done by the electric field
on a particle of charge
and see
for yourself whether that work appears to be trajectory-independent. Recall that the force
acting on a charged particle in an electric field is given by
.
Recall that the work
done on an object by a constant force is
,
where
is the magnitude of the force acting on the object,
is the magnitude of the
displacement that the object undergoes, and
is the angle between the vectors
and
.
Consider a uniform electric field
and a rectangle ABCD, as shown in the figure. Sides
AB and CD are parallel to
and have length
; let
be angle BAC.
Part A
Calculate the work
done by the electrostatic force on a particle of charge
as it moves from A to B.
Express your answer in terms of some or all the variables
, ,
, and
.
The angle
between the force and the displacement is zero here, so
, and the general formula for work becomes
.
Hint A.1
Find the angle
With reference to the given expression for the work done by a constant force, what value of
should you use here?
ANSWER:
Correct
0
ANSWER:
=
Correct
Part B
Calculate the work
done by the electrostatic force on the charged particle as it moves from B to C.
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