Scalar Potential in Electricity and Magnetism

PHY 042: Electricity and Magnetism Scalar Potential Prof. Hugo Beauchemin 1

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 E- field 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 2

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 (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 q is conservative and not dissipative The Helmholtz theorem therefore tells that the electric field can be interpreted as the variation of a scalar potential c) 3

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 4

The Scalar Potential II It has a simple geometric interpretation: The force acting on a test charge q 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 5

Advantages of V (I) There are many conceptual and empirical advantages of using the potential rather than the E-field to discuss electrostatic phenomena 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 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 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 6

Advantages of V (II) 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 Only partial information on the system is needed to make predictions on it V allows us to make predictions for experiments that we couldn t make otherwise 7

Advantages of V (III) 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 muchstronger advantage Modern physics uses potentials because requiring gauge invariance generates interactions (paradigm of particle physics) V has measurable physics effects: Aharonov-Bohm effect We can experimentally show that measurements know V 0 when E=0 8

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 9

Solutions to Poissons 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 from V, but by inverting it (solving the diff. equation) we can find V from . 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 or 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 10

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 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 11

Summary II and V 12

Boundary conditions We said that the knowledge of the particular boundary conditions is sufficient (rather than the full knowledge of the charge distribution ) 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? 13

General boundary conditions Consider an infinitesimal element on an arbitrary surface with charge density Apply Gauss law: If S 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 14