Magnetostatics 1 Introduction Charges at rest produce electric fields. When charges move with constant velocity, a static magnetic field or a magnetostatic field is produced. Thus, a magnetostatic field is produced by a constant current flow, which could be due to the magnetization currents in a permanent magnet, electron beam current in vacuum tubes or conduction currents in a current carrying conductor. The strength of the magnetic field intensity is represented → − → − → − as H (analogous to E ) and the magnetic flux density is represented as B (analogous to → − D ). 2 Biot-Savart’s Law and Magnetic Fields Just like Coulomb’s law that deals with the force exerted by a point charge, Biot-Savart’s law deals with the force exerted by a differential current element. The law states that −→ the differential magnetic field intensity, dH, produced at a point P due to a differential → − current element I dl at a distance r is given by → − − −→ I dl × → r dH = 3 4πr where × represents the cross product. Figure 1: Magnetic field at point P due to a current carrying conductor Thus, the total magnetic field due to a line current distribution is given by − − Z → → − I dl × → r H = 3 4πr L 1 3 Ampere’s Law → − Ampere’s circuit law states that the line integral of H around a closed path is the same as the net current Ienclosed enclosed by the path. Thus, I − → − → H . dl = Ienclosed L Ampere’s law in magnetostatics is very similar to Gauss’ law in electrostatics. The above equation can be modified as follows, I Z − → → − → → − − H . dl = J .dS L S H → R − → − − → where J is the current density. Now if we apply stokes law so that L H . dl = S (∇ × → → − − H ).dS, Z Z → → → − − → − − (∇ × H ).dS = J .dS S 4 S → − → − → − ⇒ ∇×H = J Magnetic Flux Density → − → − Similar to D in electrostatics, there exist an analogous magnetic flux density B in mag→ − → − netostatics. B is related to H according to → − → − B = µ0 H where µ0 is known as the permeability of free space. Its value is 4π × 10−7 H/m (Henry per metre). The magnetic flux through an surface S is given by Z → → − − B .dS ψ= S The unit of magnetic flux is Weber (Wb) and that of magnetic flux density is Wb/m2 or Tesla (T). We know that magnetic monopoles don’t exist. Thus the total magnetic flux through any closed surface S will always be zero. That is I → → − − B .dS = 0 S Figure 2: Net flux through the blue surface is 0 2 Now if we apply divergence theorem, we have I → − → − ∇. B dV = 0 V → − → − ∇. B = 0 → − → − Alternatively, if ∇. B = 0 is accepted as a fundamental postulate of magnetostatics then it can be proved that magnetic monopoles do not exist. 5 Magnetic Force As in case of electric fields, the force experience by a moving charge in a magnetic field is given by → − → − − F = q(→ v × B) → − − where → v is the velocity of the charge particle and B is the magnetic flux density. We can rewrite the equation as I → − → → − − F = I dl × B L where I is the current in a current carrying inductor whose path is given by L. Note that the magnetic field produced by the element does not exert a force on itself (similar to the → − fact that a point charge does not exert a force on itself). The B in the above equation is due to another current carrying element. 3