Download Lecture 2: Gauss' Laws and Maxwell Equations in Electromagnetics - Prof. Aurangzeb Khan and more Study notes Guiding Electromagnetic Systems in PDF only on Docsity! 1 EE 354 Electromagnetics Lecture 2 Line Integrals Surface Integrals Maxwell Equations in Integral Form: Gaussās Law for electric field Gaussās Law for magnetic field Faradayās law Ampereās circuit law Maxwell's Equations : I. Gauss' law for electricity II. Gauss' law for magnetism III. Faraday's law of induction IV. Ampere's law Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The magnetic field B is defined in terms of force on moving charge in the Lorentz force law. The interaction of magnetic field with charge leads to many practical applications. Magnetic field sources are essentially dipolar in nature, having a north and south magnetic pole. The SI unit for magnetic field is the Tesla, which can be seen from the magnetic part of the Lorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb x meter). A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss). Magnetic Field 2 The magnetic field B is defined from the Lorentz Force Law, and specifically from the magnetic force on a moving charge: The implications of this expression include: 1. The force is perpendicular to both the velocity v of the charge q and the magnetic field B. 2. The magnitude of the force is F = qvB sinĪø where Īø is the angle < 180 degrees between the velocity and the magnetic field. This implies that the magnetic force on a stationary charge or a charge moving parallel to the magnetic field is zero. 3. The direction of the force is given by the right hand rule. The force relationship above is in the form of a vector product. Magnetic Force Both the electric field and magnetic field can be defined from the Lorentz force law: The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule. Lorentz Force Law Magnetic Flux Magnetic flux is the product of the average magnetic field times the perpendicular area that it penetrates. It is a quantity of convenience in the statement of Faraday's Law and in the discussion of objects like transformers and solenoids. In the case of an electric generator where the magnetic field penetrates a rotating coil, the area used in defining the flux is the projection of the coil area onto the plane perpendicular to the magnetic field. And also magnetic flux linking the surface S is defined as the total magnetic flux density passing through S, or ā« ā
=Ī¦ S dsB Gauss' Law for Magnetism The net magnetic flux out of any closed surface is zero. This amounts to a statement about the sources of magnetic field. For a magnetic dipole, any closed surface the magnetic flux directed inward toward the south pole will equal the flux outward from the north pole. The net flux will always be zero for dipole sources. If there were a magnetic monopole source, this would give a non-zero area integral. The divergence of a vector field is proportional to the point source density, so the form of Gauss' law for magnetic fields is then a statement that there are no magnetic monopoles. Figure Magnet before and after division Electric charges can be isolated, but magnetic poles always exist in pairs and a small bar-magnet is effectively a magnetic dipole. Because magnetic field lines are continuous loops, all closed surfaces have as many magnetic field lines going in as coming out. Hence, the net magnetic flux through a closed surface is zero
