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Mathematics and Waves

This article explores the fundamental concepts of electromagnetic waves and their relation to electric and magnetic fields. It introduces key principles including the Lorentz force, Gauss's laws, Faraday’s law of induction, and Ampere’s circuit law. These laws describe how electric charge and current influence electromagnetic fields and how changing fields induce electric and magnetic forces. The content emphasizes the importance of Maxwell’s equations in classical electrodynamics and their profound impact on modern technology and the theory of relativity.

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Mathematics and Waves

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Presentation Transcript

  1. Mathematics and Waves

  2. Waves from the Sun

  3. Electromagnetic Wave

  4. Electric field – The electric field E at a point is defined as the force per unit charge experienced by a small positive test charge placed at that point. • E = F/q

  5. Magnetic field – Magnetic field B is also defined in terms of force, the charge q must be moving in order for magnetic forces to be acting. • F = q v x B

  6. Lorentz force • In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields. • F = q E + q v x B • F = [E + v x B] q

  7. Maxwell’s equations • Maxwell's equations are a set of four partial differential equations describing how the electric and magnetic fields relate to their sources, charge density and current density, and how they develop with time. • Gauss’s Law for Electric Fields • Gauss’s Law for Magnetic Fields • Faraday’s Law of Induction • Ampere’s Circuit Law

  8. Gauss’s Law for Electric Fields • The outward electric flux integrated over a closed surface is proportional to the net electrical charge enclosed by the surface. • Outward flux is defined so that field lines leaving the closed surface contribute positively while field lines entering the closed surface contribute negatively.

  9. Application • Gauss's law is very helpful in calculating electric field in those cases where electric field is symmetrical around the source producing it. Eq- spherical, cylindrical, or planer symmetry.

  10. Gauss’s Law for Magnetic field • This is the outward magnetic flux integrated over a closed surface is zero.

  11. Faraday’s Law of Induction • A changing magnetic field induces an electric field. For a curve C that bounds an area A. • Faraday's law of induction is a basic law of electromagnetism relating to the operating principles of transformers, inductors, and many types of electrical motors and generators.

  12. Example • A cylindrical region of space of radius R contains a uniform magnetic field B with direction into the page, as in fig. If the magnitude changes in time, describe the induced electric field for points inside the cylinder.

  13. Ampere’s Circuit Law • A changing electric field induces a magnetic field.

  14. Conclusion • Together with the Lorentz force law, these equations form the foundation of classical electrodynamics, classical optics, and electric circuits. • These in turn underlie the present radio, television, phone, generators, information-technologies etc. • The Maxwell equations have also been the starting point for the development of relativity theory by Albert Einstein because they predict the existence of a fixed speed of light, independent of the speed of the observer.

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