Week 7 Lex 1: Magnetostatics in free space-I

Having studied the force on a static charge due to a static chrage distribution
elsewhere (source) which was the fundamental problem of Electrostatics we now turn to studying the forces due to charges moving steadily on each other. Steadily moving charges constitute steady currents and Magnetostatics studies the forces affecting such currents. Two such current carrying conductors exert a force/unit length on each other whose direction is dependent on the direction of the steady current I. These forces are not electrostatic forces as current carrying conductors are overall neutral.


Just as in electrostatics, the fundamental problem of magnetostatics is to find the force on a test current placed at the field point due to an arbitrary current distribution elsewhere ( source currents). The starting point of magnetostatics is the Lorentz force law which is a fundamental law the force on a charge q moving with a velocity v in a magnetic field B is given as F=q v X B . For an electric field also present the force is as we know to be F=qE and this must be added to the previous expression. Moreover as this force is always perpendicular to the velocity v, magnetic forces do not do any work. It is now simple to integrate this force over the length of the conductor to find out the total force on a conductor carrying a steady current I to be F=I∫ dl X B


This is the case for what we call as a line current. Of course just like in electrostatics we can move surface charges in steady motion constitute a steady surface current density K which is the current flowing/ unit length perpendicular to the direction of flow. The magnetic force in this case is given as F=∫ ( K X B).da where the integral is now taken over a surface S. We can similarly define a volume current density J which is the current/unit area oriented perpendicular to the direction of flow. The magnetic force is now given as F=∫ ( J X B).dτ where the integral now must be taken over a volume τ.


As ∫ J.da= over a CLOSED surface S is the total current through the surface this can be related to the total change of charge density with time inside the volume enclosed by the surface S because total charge is always conserved. Using the divergence theorem we arrive at the continuity equations which states that ∇ .J is equal to the -ve of the rate of change of charge density ρ with time t. For steady currents, the charge density is constant in time as steady flow of charges should not have charge piling up anywhere. So for steady currents Div J=0


The analog of the Coulombs law for Magnetostatics is the Biot Savarts Law for magnetic fields due to steady currents. Please note that the fundamental law of Magnetism is the Lorenz force law and Biot Savart Law is a consequence of the Lorentz Force Law. Just as in coulombs law the magnetic field B due to a steady current is given by the integral of the cross product of the current vector
I=I dl'cap with the unit separation vector from the source point to the field point divided by the square of the separation vector.
The generalization to surface currents K and volume currents J is as usual.
Please check Eg 5.1 to 5.6 in Griffiths


Its easy to see that the magnetic field of an infinite staright line current circulates around the conductor according to the right hand rule in an azimuthal ( plus/minus  φ cap ) direction. The Curl of B can easily be calculated by seeing that the circulation of B over a loop C is ∫ B.dl=μ₀ I_enc where I is the current ENCLOSED by the loop C. This called Amperes law and is the analog of the Gauss Law for Electrostatics. Now using Stokes theorem its easy to see that 
∇ X B =μ₀ J where J is the volume current density. For surface currents J is to be replaced by K.


The divergence of B is ∇. B=0 . This is unlike electrostatics where Div E is equal to the charge density/epsilon zero. There are no free magnetic charges and the magnetic field lines are always CLOSED. This can be easily proved for a volume current density J using similar techniques to those in electrostatics. ( Refer to Griffiths for the derivation where like before they show that a surface contribution from the boundary is ZERO.)


Please read Section 5.34 Griffiths for a comparison of Electrostatics and Magnetostatics. Also look through the Eg. 5.7-5.10 Griffiths.