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PHY 042: Electricity and Magnetism
Scalar Potential
Prof. Hugo Beauchemin
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Introduction
 So far we defined the E-field, generalized the Coulomb’s force to
Gauss’ law, and used the flux-divergence and Stokes theorems to get
and
 The Helmholtz’ theorem then guaranteed that with the knowledge
of the proper boundary conditions, we are able to predict the Efield everywhere, in any electrostatic configurations and therefore
for any electrostatic experiments

We didn’t fully exploited this yet, but will comeback to this later
 The electric field is a fundamental concept, with well-defined
empirical content and meaning, which generalizes Coulomb’s law
 We will once again use the Helmholtz’ theorem to introduce
another rich and important concept: the scalar potential
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A very special field
 The electric field is a vector field but NOT any vector field because it
has to satisfy some specific field equations

E.g.: There is no r(x,y,z) such that
 The fact that the curl of an electrostatic field is null contains a lot of
information. It tells you:
a) The system is static
 Inherited from the experimental conditions in which Coulomb’s law
has been obtained
b) The force generated by E on dq is conservative and not dissipative
c)
The Helmholtz’ theorem therefore tells that the electric field can
be interpreted as the variation of a scalar potential
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The Scalar Potential I
 We have a set of 2 equations:


Gives its empirical physics content to the electric field
Gives the statics, conservative and scalar potential
conditions that have to be satisfied by the electric field
All this information can be summarized into one scalar quantity:
V(x,y,z)
 Dealing with V means dealing only with the independent
component of the E-field, which can simply be obtained from V by
application of the constrains

The E-field flows in direction of biggest changes of V
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The Scalar Potential II
 It has a simple geometric interpretation:
The force acting on a test charge dq tends to bring this charge
to the state of lowest potential as quickly as possible

Like a free falling object tries to reach the lowest gravitational
potential state as quickly as possible
 There is a connection to the known concept of potential energy
The potential (V) and the potential energy (U) are completely
different concepts but share a connection in that minimizing
the potential also minimizes the potential energy, which gets
converted in kinetic energy
 Will be exploited later
 We can easily draw equipotential lines:
Surface of constant potential
obtained by solving V(x,y,z)=C
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Advantages of V (I)
 There are many conceptual and empirical advantages of using the
potential rather than the E-field to discuss electrostatic phenomena
① This quantity is easily controllable in experiments, using batteries for
example. It is thus the concept that has the most direct empirical
meaning, and can be used to provide an empirical meaning to other
concepts in contexts where Coulomb’s law wouldn’t apply
② It can be used to develop empirical and pragmatic rules that
wouldn’t be formalized otherwise

E.g.: V=RI
 This is a steady current situation described by V… not in electrostatics
③ It simplifies the problems to be solved because there is no need to
deal with vectors and constraints anymore. One simply need to deal
with the independent element of E summarized in V

Note that some geometry still makes it easier to work with E
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Advantages of V (II)
④ With the potential, we have one simple differential equation, the
Poisson’s equation. By solving it, we can determine E everywhere:
 Outside of the charge distribution, this equation is still meaningful:
it is the Laplace’s equation:
 It allows to find E everywhere without the need to know r
 Only partial information on the system is needed to make
predictions on it
⑤ V allows us to make predictions for experiments that we couldn’t
make otherwise
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Advantages of V (III)
⑥ V is uniquely defined up to a choice of gauge
The electric field is independent of constants added to V
 Only care about difference of potential, and not about absolute values

 In order to define V(r) from E(r), we need to fix the value of V at
some reference point
 Choose a gauge
⇒ This has the advantage of allowing for the simplification of some
problems by making a suitable choice of reference, but there is a
much stronger advantage…
Modern physics uses potentials because requiring gauge
invariance generates interactions (paradigm of particle physics)
⑦ V has measurable physics effects: Aharonov-Bohm effect

We can experimentally show that measurements know V≠0 when E=0
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Extra notes on V
 Q: Why E is defined as “-” grad(E)?
A: To ensure that V>0 if q>0
 The superposition principle, obtained from Coulomb’s law and
“transmitted” to the electric field, ALSO applies to the potential

Things are easier since we are dealing with a scalar sum
 A new unit is defined for the potential: the Volt
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Solutions to Poisson’s eqn
 In the examples seen in class, we used E to compute V, but the
whole point of introducing V was the opposite…
 We can use the Poisson’s equation to do this

The Poisson’s equation tells you how to get r from V, but by
inverting it (solving the diff. equation) we can find V from r.
 We don’t know yet how to directly solve the Poisson’s equation,
but we can use some known examples to find a set of solutions
satisfying the equation
Equivalent formula exist for
l or r charge densities
 These solutions are only valid for a choice of gauge V(∞)=0. For
an infinite rod, the potential would diverge
 Victim of ideal simplifications of edge effects…
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Summary I
 We started from simple experimental considerations…
 Static cases at equilibrium
 The Coulomb’s experimental setup
 … and with mathematics and physics principles
 Linearity and superposition principle
 We introduced three concepts allowing us to generalize the physics
content of Coulomb’s law to any experiments involving
electrostatic field
The electric field E
 The electric scalar potential V
 The charge density r

Each has a physics empirical meaning,
and advantages regarding applications
depending on the information available
 We used math tools generalizing the relationships between these
concepts
Gauss’ theorem
 Helmholtz’ theorem

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Summary II
r
and
V
E
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Boundary conditions
 We said that the knowledge of the particular boundary conditions is
sufficient (rather than the full knowledge of the charge distribution
r) to find the potential V in a large number of relevant situations.
Q: Can we make general statements about what boundary conditions
are or it is completely system-dependent?
 E.g.: What can we say about the electric field of a combination of
infinite planes?

E is completely determined by  0

E is discontinuous at the position of each planes, i.e. at the various
boundaries of the charge distributions
 Can we generalize this to any system configuration?
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General boundary conditions
 Consider an infinitesimal element on an arbitrary surface with
charge density 
 Apply Gauss’ law:
If DS is small enough,
the charge distribution
looks like an ∞ plane
 Decompose E in normal and
tangent components
and solve for both
 Find a general set of boundary conditions that always apply:
Discontinuity of
the normal
component
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