Electromagnetism

1. Electromagnetism dS V We want to apply the reaction theory developed in the first few lectures to electronuclear interactions. It is worthwhile reviewing some facts from electricity and magnetism. CONSERVATION of CHARGE J S We used the divergence theorem to convert the integral over the surface S into an integral over the volume V. We assume that the volume doesn’t change, so only the charge density r in the volume changes. This leads to the continuity equation.

2. ELECTROSTATICS Coulomb’s Law: This is a fundamental law of electrostatics. We will use the Gaussian system of units here. This is also measured in cgs. The unit of charge is the stat-coulomb. This amount of charge exerts a force of one dyne on an equal charge located 1 centimeter away. The electric field is a vector field which for a point charge is defined by E r q From this expression for the electric field we derive Gauss’ law. We can convert the integral expression into a differential law,

3. MAGNETOSTATICS This case means that currents are constant. In particular the charge density r(r) is independent of time. The consequence of this is that the divergence of the current density J is zero. This is a result we can deduce from the equation of continuity ( eqn 1). The fundamental experimental law in magnetostatics is the Biot-Savart Law. The contribution dB, to the magnetic field, B, at point P, due to the current I in a short length dl is given by. P r I dl JdS We consider a current density J into an area dS. We can rewrite (5) using

4. Eqn. 6 can be written in terms of an auxiliary potential A, called the vector potential. Notice that the addition of the gradient of an arbitrary function y in eqn 7 does not change the result we get for B in eqn 6. This arbitrariness in the definition of the vector potential A can be used to advantage to simplify equations. This flexibility is called “gauge invariance”. Consequently we write for B, We can find another relation between B and J from Ampere’s Law. C is the path encircling a surface S. I is the total current through surface S. We used Stokes theorem to connect the first and third integral.

5. So for magnetostatics we get an alternate expression for Ampere’s law, ELECTRODYNAMICS Now consider the case where currents change in time. The charge density is now a function of time too. FARADAY’S LAW The EMF generated around the curve C depends on the changing magnetic flux through area S. B B C S E dl Take the case where the geometry remains constant so that only B changes in time. Then using eqn 8,

6. From eqn 10 we note that an induced electric field arises from changing vector potentials. However, we also know from electrostatics that the electric field is also generated by a gradient of an electric potential. We take into account both of these sources of the electric field by writing, In eqn (12) we used that fact that the curl of a gradient is zero. In the case of the magnetic field we can see that Ampere’s law can not be the whole story. It can not only be a current which gives rise to the magnetic field. B=0 ? B B C I I

7. Consider the case of charging up a capacitor C which is connected to very long wires. The charging current is I. From the symmetry it is easy to see that an application of Ampere’s law will produce B fields which go in circles around the wire and whose magnitude is B(r) = 2I/(cR). But there is no charge flow in the gap across the capacitor plates and according to Ampere’s law the B field in the plane parallel to the capacitor plates and going through the capacitor gap should be zero! This is clearly unphysical. We also note from the differential form for Ampere’s law, eqn (9) that But the divergence of J is zero only for magnetostatics. Maxwell saw that Ampere’s Law, in the form of eqn 9, needed to be to modified to included an additional current-like term, JD, called the displacement current. We now use Gauss’ Law in the form of eqn 3.

8. MAXWELL’S EQUATIONS for the VACUUM , Continuity equation

9. ALTERNATE FORM OF MAXWELL’S EQUATIONS FROM THE POTENTIALS From eqn’s 11 and 8 we see that we can write the electric and magnetic fields E and B in terms of the scalar potential f and the vector potential A. Substituting into Maxwell’s equations we obtain, Because of gauge invariance referred to in eqn 7, we have some flexibility in our choice of A. In order that both B and E remain unchanged we must make the following changes to the scalar potential if the vector potential is modified. This flexibility in choice of A allows us to choose y in such a way that

10. Using eqn 15 the equations for the scalar and vector potentials become Eqns 16, 17, and 18 form an equivalent set of Maxwell’s equations. The choice of gauge which yields eqn 16 is called the Lorenz gauge. This gauge is particularly useful when the explicit Lorentz invariance of the theory is needed. Another popular choice of gauge called the Coulomb gauge, is used mainly in low energy situations, such as atomic physics. The Coulomb gauge yields a Poisson equation for the electric potential.

11. REFERENCES • “Classical Electrodynamics”, 2nd Edition, John David Jackson, John Wiley and Sons, 1975 • “Electrodynamics”, Fulvio Melia, University of Chicago Press, 2001 • “Relativistic Quantum Mechanics and Field Theory”, Franz Gross, John Wiley and Sons, 1993