Electric Potential Energy
We can store gravitational potential energy (UG) in a mass by moving
it against the force exerted by a uniform gravitational field…
UG = U0 + (mg)h
(where U0 is the potential energy value we assign at h = 0).
Likewise, we can store electric potential energy (UE) in a charge by
moving it against the force exerted by a uniform electrical field…
UE = U0 + (qE)d
(where U0 is the potential energy value we assign at d = 0).
For either object, if we then release it to move freely in response to
the field’s force, its potential energy is reduced; it “falls.”
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OSU PH 213, Before Class #10
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It’s in this simplest possible electrical field—a constant field of the kind
you would find between the plates of a parallel capacitor—that the analogy
between a gravitational field and an electrical field is closest:
The distance you would “lift” a charge toward the plate matching its sign is
much like the distance you would lift a mass against the force of gravity.
Either of these “lifts” would increase the object’s potential energy.
And then if you were to allow the field to lower the object (i.e. let it “fall”),
the potential energy would be reduced.
So if a charge q is somewhere in an electrical field, it has a certain electrical
potential energy, UE. That is, if you place it there and release it, it will
“fall”—either in the same direction as the field lines (if it’s positive) or in
the opposite direction (if it’s negative).
4/24/17
OSU PH 213, Before Class #10
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Electric Potential Energy (UE)
and Electric Potential (“voltage”) (V)
In the study and harnessing of electrical energy, it is convenient to
express the electrical potential energy on a per-unit-charge basis.
This is called the electric potential or voltage, and is denoted by V.
Electric potential is a field—a point-by-point description of space—
but it’s an energy field, not a force field.
If a charge q is at a point in space that has an electric potential value
V, then its electrical potential energy is given by UE = qV
The units of V are Joules/Coulomb (also known as Volts). Note that
we speak of electric potential—like potential energy—in terms of its
changes.
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OSU PH 213, Before Class #10
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So we can now express—in the form of energy-per-Coulomb—the
difference between any two points in space where we could place a
charge: DUE/q = (UE.B – UE.A)/q = VB – VA = DV
Stated most simply:
DUE = qDV
In the simplest case—in a uniform electrical field—comparing the
potential energy-per-Coulomb stored in the same charge at two
different places (A and B), we have this:
UE.B = U0 + (qE)dB
UE.A = U0 + (qE)dA
DV = (UE.B – UE.A)/q = (qEdB – qEdA)/q = EDd
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OSU PH 213, Before Class #10
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Again: DUE = qDV
Notice also that the sign of the charge q is useful and important:
If q is a positive charge, then it loses UE when it moves from a point
of higher voltage to lower (and gains UE going the other way).
If q is a negative charge, then it gains UE when it moves from a point
of higher voltage to lower (and loses UE going the other way).
In other words: A positive charge “falls” from higher voltage
toward lower. A negative charge “falls” from lower voltage toward
higher.
4/24/17
OSU PH 213, Before Class #10
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The Electric Potential Energy of Collections of Point Charges
Q: Uniform electrical fields between parallel charged plates are created by
millions of charges, all crowded together. But what about when the fields
are not uniform—such as when just a few point charges create them? What
work does it take to move one point charge, q2, in the vicinity of another q1?
A: Again, we look to gravitation for analogy, but this time, instead of
assuming a constant local E, we use the general case (E = kq1/r2):
FE.12 = kq1q2/r2 in the r-direction (where the r-origin is at q1)
Integrating this over the distance r:
UE = kq1q2/r
Notice: UE = 0 when r = ∞ (just like UG). And UE is negative when the
force between the charges is attraction (just like UG), but it’s positive when
the force is repulsion. (This makes sense: It takes positive work to bring
two like charges closer together.)
4/24/17
OSU PH 213, Before Class #10
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The Strength of the Electric Potential Field
(Voltage) Created by a Point Charge
Q: Again: DUE = qDV So, what is the electric potential (voltage)
at any point in space due to a nearby point charge, q?
A: Dividing UE.point.charge by q, we get the result immediately:
V = kq/r, where r is the distance from q to the point in question.
Note where we have selected V = zero only at an infinite distance
from q. (The analogy to gravity again.)
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OSU PH 213, Before Class #10
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When a Charge Moves without a Change in Voltage
Finally, this question to start mulling over….
Q: Suppose two point charges (A and B) are nearby one another.
What path could charge A follow so that its UE would not change?
(Hint: What if charge A were in a uniform field instead of a field
created by a point charge?)
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OSU PH 213, Before Class #10
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