Maxwell’s Equations: Maxwell’s Equations Presented by : Debjyoti Banerjee Anshuman Rai Anmol Anupam Ashita Bhatnagar
Maxwell’s Equations: Maxwell’s Equations We have been examining a variety of electrical and magnetic phenomena James Clerk Maxwell summarized all of electricity and magnetism in just four equations Remarkably, the equations predict the existence of electromagnetic waves
Maxwell’s Equations: Maxwell’s Equations As you can see we need some vector calculus to use these equations. That isn’t going to happen in this class, but I wanted you to see the equations anyway.
Maxwell’s Equations: Maxwell’s Equations The first is Gauss’s Law which is an extended form of Coulomb’s Law The second is the equivalent for magnetic fields, except that we know that magnetic poles always occur in pairs (north & south)
Maxwell’s Equations: Maxwell’s Equations The third is Faraday’s Law that a changing magnetic field produces an electric field The fourth is that a changing electric field produces a magnetic field The latter is a bit of a stretch. We knew that a current produces a magnetic field
Maxwell’s Equations: Maxwell’s Equations Start with Ampere’s Law Earlier, we just went on a closed path enclosing surface 1. But according to Ampere’s Law, we could have considered surface 2. The current enclosed is the same as for surface 1. We can say that the current flowing into any volume must equal that coming out.
Maxwell’s Equations: Maxwell’s Equations Suppose we have a charged capacitor and it begins to discharge Surface 1 works but surface 2 has no current passing through the surface yet there is a magnetic field inside the surface.
Maxwell’s Equations: Maxwell’s Equations Same problem here. Surface 1 works, but no current passes through surface two which encloses a magnetic field. What is happening???
Maxwell’s Equations: Maxwell’s Equations While the capacitor is discharging, a current flows The electric field between the plates of the capacitor is decreasing as current flows Maxwell said the changing electric field is equivalent to a current He called it the displacement current
PowerPoint Presentation: Inconsistency of Ampere’s Law: Taking divergence of Ampere’s law hence Ampere’s law is not consistent for time varying equation of continuity. (from Gauss’s Law) displacement current density. Hence Ampere’s law becomes Now taking divergence results equation of continuity Integrating over surface and applying Stokes’s theorem magneto motive force around a closed path=total current enclosed by the path.
Maxwell’s Equations: Maxwell’s Equations
Electromagnetic Waves: Electromagnetic Waves So, a magnetic field will be produced in space if there is a changing electric field But, this magnetic field is changing since the electric field is changing A changing magnetic field produces an electric field that is also changing We have a self-perpetuating system
Electromagnetic Waves: Electromagnetic Waves Close switch and current flows briefly. Sets up electric field. Current flow sets up magnetic field as little circles around the wires. Fields not instantaneous, but form in time. Energy is stored in fields and cannot move infinitely fast.
Electromagnetic Waves: Electromagnetic Waves Picture a shows first half cycle. When current reverses in picture b, the fields reverse. See the first disturbance moving outward. These are the electromagnetic waves.
Electromagnetic Waves: Electromagnetic Waves Notice that the electric and magnetic fields are at right angles to one another! They are also perpendicular to the direction of motion of the wave.
Speed of EM Waves: Speed of EM Waves
PowerPoint Presentation: Word statement of field equation: The magneto motive force (magnetic voltage) around a closed path is equal to the conduction current plus the time derivative of electric displacement through any surface bounded by the path. The electromotive force (electric voltage)around a closed path is equal to the time derivative of magnetic displacement through any surface bounded by the path Total electric displacement through the surface enclosing a volume is equal to the total charge within the volume. The net magnetic flux emerging through any closed surface is zero.
Thank you: Thank you