Download Electric Potential Electric Potential Energy versus Electric Potential

Electric Potential
Electric Potential Energy versus Electric Potential
Gravitational Force and Potential Energy
First we review the force and potential energy of an object of mass
gravitational field that points downward (in the
near the earth's surface.
Part A
Find the force
height
on an object of mass
in a uniform
direction), like the gravitational field
in the uniform gravitational field when it is at
.
Express
in terms of
, , , and .
=-m*g*z_unitCorrect
Because we are in a uniform field, the force does not depend on the object's location.
Therefore, the variable does not appeaer in the correct answer.
Part B
Now find the gravitational potential energy
height . Take zero potential to be at position
of the object when it is at an arbitrary
.
Express
in terms of , , and . Note that because potential energy is a scalar, and
not a vector, there will be no unit vector in the answer.
=m*g*zCorrect
Part C
In what direction does the object accelerate when released with initial velocity upward?
downward
Correct
Electric Force and Potential Energy
Now consider the analogous case of a particle with charge placed in a uniform electric
field of strength
Part D
Find
, pointing downward (in the
direction)
, the electric force on the charged particle at height .
Express
in terms of ,
, , and .
=-q*E*z_unitCorrect
Part E
Now find the potential energy
of this charged particle when it is at height . Take
zero potential to be at position
Express
.
(a scalar quantity) in terms of ,
, and .
=q*E*zCorrect
Part F
In what direction does the charged particle accelerate when released with upward initial
velocity?
upward or downward depending on its charge
Correct
Electric Field and Electric Potential
The electric potential is defined by the relationship
potential energy of a particle with charge .
Part G
, where
is the electric
Find the electric potential of the uniform electric field
. Note that this field is
not pointing in the same direction as the field in the previous section of this problem.
Take zero potential to be at position
Express
in terms of ,
.
, and .
=-E*zCorrect
Part H
The SI unit for electric potential is the volt. The volt is a derived unit, which means that it
can be written in terms of other SI units. What are the dimensions of the volt in terms of
the fundamental SI units?
Express your answer in terms of the standard abbreviations for the fundamental SI units:
(meters), (kilograms), (seconds), and
volts =kg*m^2/(s^2*C)Correct
(coulombs)
Part I
The electric field can be derived from the electric potential, just as the electrostatic force
can be determined from the electric potential energy. The relationship between electric
field and electric potential is
, where
is the gradient operator:
.
The partial derivative
variables constant.
means the derivative of
Consider again the electric potential
corresponding to the field
potential depends on the z coordinate only, so
Find an expression for the electric field
with respect to , holding all other
. This
and
in terms of the derivative of
.
.
Express your answer as a vector in terms of the unit vectors , , and/or . Use
the derivative of
for
with respect to .
=-dV/dz * z_unitCorrect
Part J
A positive test charge will accelerate toward regions of ________ electric potential and
________ electric potential energy.
Choose the appropriate answer combination to fill in the blanks correctly.
lower; lowerCorrect
Part K
A negative test charge will accelerate toward regions of ________ electric potential and
________ electric potential energy.
Choose the appropriate answer combination to fill in the blanks correctly.
higher; lower
Correct
A charge in an electric field will experience a force in the direction of decreasing
potential energy. Since the electric potential energy of a charge is equal to the charge
times the electric potential (
), the direction of decreasing electric potential energy
is the direction of increasing electric potential.
The Fate of an Electron in a Uniform Electric Field
First, let us review the relationship between an electric field and its associated electric
potential
Part A
. For now, ignore the electron located between the plates.
Calculate the electric potential
inside the capacitor as a function of height . Take
the potential at the bottom plate to be zero.
Express
in terms of
and .
=E*hCorrect
Now an electron of mass
and charge
(where
within the electric field (see the figure) at height
Part B
is a positive quantity) is placed
.
Calculate the electon's potential energy
, neglecting gravitational potential energy.
Express your answer in terms of
.
,
, and
=-q_e*E*h_0Correct
Part C
The electron, having been held at height
(i.e.,
, is now released from rest. Calculate its speed
) when it reaches the top plate.
Express in terms of , , , , , and other given quantities and constants.
=sqrt(2*q_e*E*(h_1-h_0)/m)Correct
Now we consider the effect of changing either the charge or the mass of the charged
particle that is released from rest at height .
Part D
Imagine a particle that has three times the mass of the electron. All other quantities given
above remain the same. What is the ratio of the kinetic energy
that this heavy
particle would have when it reaches the upper plate to the kinetic energy
electron would have? That is, what is
1
Correct
that the
?
Part E
Imagine a third particle, which we will call a cyberon. It has three times the mass of an
electron ( ). It has a positive charge that is three times the magnitude ( ) of the
charge on an electron. What is the ratio of the speed that the cyberon would have when
it reaches the upper plate after being released from rest at position
to the speed
that
the electron would have? That is, what is ?
none of the above: the cyberon will never reach the upper plateCorrect
Because it has positive instead of negative charge, the cyberon will accelerate downward,
toward the lower plate.
Energy Stored in a Charge Configuration
Part A
What is , the amount of work it took to assemble this charge configuration if the point
charges were initially infinitely far apart?
Express your answer in terms of some or all of the variables , , and .
=0Correct
The hints led you through the problem by adding one charge at a time. A little thought
shows that this is equivalent to simply adding the energies of all possible pairs:
.
Note that this is not equivalent to adding the potential energies of each charge. Adding
the potential energies will give you double the correct answer because you will be
counting each charge twice.
Part B
Which of the following figures depicts a charge configuration that requires less work to
assemble than the configuration in the problem introduction? Assume that all charges
have the same magnitude .
figure a
Correct
Charged Mercury Droplets
Part A
Find
, the ratio of
, the electric potential of the initial drop, to
potential of one of the smaller drops.
The ratio should be dimensionless and should depend only on
, the electric
=n^(2/3)Correct
Potential of a Finite Rod
Part A
What is , the electric potential at point A (see the figure), located a distance above
the midpoint of the rod on the y axis?
Express your answer in terms of , , , and .
=k*q/L*ln((sqrt(L^2/4+d^2)+L/2)/(sqrt(L^2/4+d^2)-L/2))Correct
If
, this answer can be approximated as
.
For
,
. For this problem, this means that the logarithm can be further
approximated as
, and the expression for potential reduces to
This is what we expect, because it means that from far away, the potential due to the
charged rod looks like that due to a point charge.
.
Part B
What is , the electric potential at point
(on the x axis)?
, located at distance from one end of the rod
Give your answer in terms of , , , and .
=(k*q*ln((d+L)/d))/LCorrect
This result can be written as
.
As before, for
,
. Thus, for
, the logarithm approaches
, in
which case the result reduces to . This is what we expect, because it means that from
far away, the potential due to the charged rod looks like that due to a point charge.