Download Electric potential energy and electric potential . Today we wax

Electric potential energy and electric potential . Today we wax mathematical for awhile, but stick
with me, in the end we will have a few new powerful concepts and simple formulas to work with.
Work and kinetic energy. Recall from mechanics the formula
that we derived from the equations of motion for constant acceleration
which arise from a constant force acting along (parallel to) the line of motion.
We obtain the formula by solving for time in the second equation
and substituting this value of time into the first equation to eliminate the time variable.
Ever wonder where we get our definitions of work and kinetic energy? Multiply the formula by
Recognize anything? The kinetic energy and the work terms!
The formula is none other than the work-energy theorem in disguise.
What if the force is not parallel to the displacement? Recall from mechanics the more general case of
work done by a constant force
acting through a displacement
, with angle
between the vectors
Gravitational potential energy. For the gravitational acceleration field
The work done by the gravitation field on a body of mass
, acting through a displacement
is
Are you wondering why I use the subscript G (we did not use it in mechanics)? Because soon we will be
using the subscript E for the electric force!
Note that the change in elevation (height) is given by
It does not matter what path the body takes, straight up or down, along an incline, or some curved path:
all that matters is the change in elevation, which is the displacement parallel to the field, thus
In mechanics, we noted something special about the gravitational field: it conserves energy. All of the
energy “stored” in the field when you lift a book is recovered as energy of motion when the book falls.
If the potential energy of a field is path independent, it is a conservative field.
We defined the change in gravitational potential energy as the negative of the work done by the
gravity field acting on a body of mass through a displacement
Here, it is worthwhile to remember that , the change in elevation, can be positive or negative,
corresponding to the body moving up or down.
If we define a zero reference (we define the potential energy to be zero at a certain elevation)
then we omit the
symbol write
and think of it as an “absolute” potential energy that everyone in the universes agrees on. There is no
such thing as an absolute potential energy, there is only relative potential between two points, but that
does not stop us from pretending we have an absolute! Here, “absolute” means simply that everyone
“agrees” on a zero reference.
Now let’s talk about the work done by a uniform the electric field
and we define the change in electrical potential energy as the negative of the work done by the electric
field on a body of charge
We take things one step further and define “electric potential”
and define a new unit
You are all familiar with volts, as in “1.5V battery”. What does that mean? If a charge of
moves from the top of the battery to the bottom (say along a wire), it experiences a change in potential
and a change in potential energy of
Supposed you attach the battery to parallel plates and let an electron “fall” upward through the
potential difference. How fast will the electron be moving? Let
If we discharge the 1.5V battery through a wire, does this mean the electrons end up moving this fast?
No, because a wire is filled with atoms, it is an obstacle course; the electron bumps into atoms all along
its way and loses energy, causing the atoms to vibrate more (heat up). By the time the electron reaches
the bottom, its tongue is hanging out, it is tired and flops onto the ground motionless, having lost all of
its energy
to heat.
What did we learn today? The electric potential , the potential field
electric field
and its relationship to the
, for two cases: 1) a uniform electric field, and 2) the electric field of a point charge.
We introduced two simple handy formulas
If you are given the voltage (the potential) at points A and B, the potential difference
can be used to find the change in potential energy experienced by a charge as it moves from A to B,
which can be used in the work-energy theorem
to analyze the motion of the charge.
Finally, we discussed how the electric field is a conservative field, which means that energy is conserved
by the field. As with the gravitational field, all of the energy “stored” in the field is given back, none is
lost to friction (we neglect electromagnetic radiation produced by the relative acceleration of charged
particles; some energy is carried away by radiation).
Exam Friday, lookup your exam location on myWPI, it is based on your section number!