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Electric Potential
AP Physics C
Conservative Forces and Energy
 Remember from our lecture on work and energy that we defined a
category of forces called conservative forces.
 If a force is conservative, we can make a few claims:
 There is a potential energy associated with the force.
 The work done by the force is path independent.
 We can express the work done by the conservative force as the
negative change in potential energy:
𝑊𝑐𝑜𝑛𝑠 = −∆𝑈
 In addition, the total mechanical energy is conserved in the absence of
non-conservative forces.
Gravitational Potential Energy
 Remember back to gravitational
potential energy. As an object falls
to the Earth, gravity does positive
work on the object.
 As the object falls, the object gains
kinetic energy and looses
potential energy, meaning that
our change in potential energy is
negative.
𝑊𝑔𝑟𝑎𝑣 = −∆𝑈𝑔𝑟𝑎𝑣
 We will make this analogous to a
charged particle moving through
an electric field.
Electric Potential Energy –
Uniform Electric Field
 Inside a parallel plate capacitor (two
oppositely charged plates) there is an
electric field present. If a positive charge is
placed between the plates, there is a force
exerted on the charge by the field. That
force then does work on the charge.
𝐹 = 𝑞𝐸
𝑊 = 𝐹𝑑𝑐𝑜𝑠 𝜃
𝑊𝑒𝑙𝑒𝑐 = 𝑞𝐸𝑑
 Because electric forces are conservative,
we can determine the formula for
electric potential energy.
𝑊 = −∆𝑈
𝑞𝐸𝑑 = −∆𝑈 → 𝑈 = 𝑈𝑜 + 𝑞𝐸𝑑
Electric Potential Energy –
Uniform Electric Field
 As charges approach the plate with opposite charge, they loose
potential energy and gain kinetic.
 This is because the particles are being accelerated!
Electric Potential Energy – Point
Charges
 Consider two positive charges. If we wish to move the charges closer to each
other, we must do work on the charges.
 Just as we did with gravity, we need to choose a reference point, and we will
again choose infinity (the potential at that point will be zero).
 Because electric forces are conservative, we can integrate coulombs law to find
the potential between these two charges.
𝑟
𝑊=−
𝑈𝑒𝑙𝑒𝑐 =
𝐹𝑑𝑟 = −
1 𝑞1 𝑞2
4𝜋𝜀𝑜 𝑟
1 𝑞1 𝑞2
𝑞1 𝑞2 −1 1
𝑑𝑟
=
−
∗
+
2
4𝜋𝜀
𝑟
4𝜋𝜀
𝑟
∞
𝑜
𝑜
∞
Electric Potential
 Often we are not concerned so
much with the electric potential
energy of a system, but rather the
electric potential. I know, it
sounds like the same thing, but
there is a slight difference.
 Electric potential (often called
potential difference or voltage) is
the amount of electric potential
energy per unit charge.
 The SI unit for potential is the
volt.
Electric Potential
 A test charge q is used as a
probe to determine the
electric potential, but the
value of V is independent of
q.
 The electric potential, like
the electric field, is a
property of the source
charges, not the test charge.
𝑉𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 =
𝑈𝑒𝑙𝑒𝑐 = 𝑞𝑡𝑒𝑠𝑡 𝑉
𝑈𝑒𝑙𝑒𝑐
1 𝑄
=
𝑞𝑡𝑒𝑠𝑡 4𝜋𝜀𝑜 𝑟
Potential and Superposition
 If you have multiple point charges, you can find the electric potential
at different points in space by calculating the potential from each
charge.
 This means that electric potential obeys the principle of
superposition!
 Electric potential is a scalar value as well, so calculations are drastically
easier than electric fields!
𝑉𝑡𝑜𝑡𝑎𝑙
1
=
4𝜋𝜀𝑜
𝑄
𝑟
Example
Calculate the electric potential at the point indicated.
Electric Potential
 Positive charges tend to move towards areas of lower potential, which
is analogous to a ball falling to the earth.
 Negative charges are the opposite, they tend to move towards higher
potentials.
 In both cases, the charge moves towards the area with an opposite
voltage, converting potential energy into kinetic.
Electric Potential in a Parallel
Plate Capacitor
 We said that the electric potential
energy of a point charge in a parallel
plate capacitor is:
𝑈 = 𝑞𝐸𝑑
 Using our idea of voltage, we can
conclude that the potential difference
between the plates in the capacitor is:
𝑉=
𝑈
= 𝐸𝑑
𝑞
 Again, notice that the potential difference is
independent of the test charge.
Equipotential Lines and Electric
Field Lines
 Any charged object emits an electric field,
and there is a potential difference associated
with it.
 We can see in these two pictures that there
are lines along which the electric potential
does not change. These are called
equipotential lines.
 You can move a charge along these lines and
no work is done, because the potential
energy does not change.
 The electric field lines are always
perpendicular to these equipotential lines.
Choice of Zero Potential
 Because only differences in potential are relevant, the choice of zero
potential is completely arbitrary.
 The three scenarios below represent the same situation, even though their
zeros are at different points.
Making Comparisons
 We are able to relate force and
potential energy (when the force is
conservative), and we can do the
same thing with electric fields and
electric potential.
𝑊=−
𝐹𝑑𝑟 → V = −
𝐸𝑑𝑟
 To find the potential difference
between charged objects, we again
choose a reference at infinity and
integrate our electric field function.
𝑟
𝑉=−
𝐸𝑑𝑟
∞
Gauss’s Law and Potential
 Often we will have problems
where you will need to find the
electric field and electric
potential as functions of
position for some charged
object.
 Gauss’s law will allow us to find
the field in all regions of space,
and that field function can be
integrated to find the potentials
at different points in space.
 We will often graph these fields
and potentials as functions of r.