Maxwell and Classical Electromagnetism – Learn


Maxwell’s Unifying Theory for Electromagnetism

James Clerk Maxwell is famous for developing his equations which explained a link between electricity, magnetism and light. His Theory on Electromagnetism provided a unifying theory that linked all the work that had previously been done on electricity and magnetism. Some of these works included:

  • Danish physicist Hans Orsted observed a magnetised compass needle deflected from its alignment to the Earth’s magnetic field when a nearby electric circuit was switched on and off. This showed that a wire carrying an electric current generates a magnetic field, and it revealed the first evidence of a relationship between electricity and magnetism.
  • English physicist Michael Faraday demonstrated in the 1830’s that changing magnetic fields produced electric fields.

Maxwell’s work quantified these relationships through a precise mathematical study of electric and magnetic effects. His work, known as Maxwell’s equations, show that electric and magnetic fields move at a speed that closely match experimental estimates of the speed of light. Maxwell went on to develop a comprehensive theory of electromagnetism which explained that light is a form of electromagnetic radiation (EMR). He also predicted that a large range of frequencies was possible for different forms of EMR beyond the visible spectrum.


Maxwell’s Equations

The equations that Maxwell developed in his theory of electromagnetism are based on vector calculus which is beyond the scope of this course and will not be analysed quantitatively here. The equations are named after physicists who played a significant role in the work that led to Maxwell developing his theory. The laws are:

  • Law 1 – Gauss’s law: This law describes the electric flux produced by electric charges.
  • Law 2 – Gauss’s law for magnetic fields: This second law is very similar to the first but applies to magnetic rather than electric fields.
  • Law 3 – Faraday’s law: This describes the electric field induced by a changing magnetic field.
  • Law 4 – Ampère-Maxwell law: This is a little like Faraday’s law but it deals with changing electric flux.

Prediction of Electromagnetic Waves

Maxwell’s theory of electromagnetism combined all the theory and observations that had been developed in relation to electrical and magnetic physics and summarised this using four equations. He also demonstrated mathematically that an electromagnetic wave was expected. Qualitatively, Maxwell’s equations summarise the interactions between electric and magnetic fields and this led to the prediction of an electromagnetic wave which can propagate through space.

He considered that if a changing electric field is produced by moving a charged particle backwards and forwards, then this changing electric field will produce a magnetic field at right angles to the original electric field. The changing magnetic field would then also produce a changing electric field and this cycle could be repeated infinitely. The result of this would be two mutually propagating fields. The electromagnetic radiation would be self-propagating and would extend outwards into space as an electromagnetic wave of a fixed frequency. Further to this, both the electric and magnetic fields would necessarily oscillate at the same frequency. The diagram below illustrates an electric field perpendicular to a magnetic field propagating through space as an electromagnetic wave. The fields are perpendicular to the direction of propagation.

Any charge that is exposed to electromagnetic radiation will respond to the electric field in the radiation and be accelerated according to F = qE. Further to this, any charge will experience a force F from a magnetic field according to F = qvBsinθ. The result of this is that electromagnetic radiation can be transformed into kinetic energy.


Prediction of Velocity

Maxwell’s calculations provided a theoretical value for the speed at which an electromagnetic wave should propagate through space. This speed so closely matched experimental values for the speed of light that it led physicists to the idea that light was a form of electromagnetic radiation. The speed of light is accepted to be 299792458 m/s. In calculations, the speed of light, c, is often accepted as 3×108 m/s.

For EMR, there is a special variation of the wave equation (v=fλ) that relates the speed of EMR/light and the frequency and wavelength of any electromagnetic wave: c=fλ.


 Example:

What is the frequency of red light which has a wavelength of 620nm?

Answer:

Using: c=f\lambda

f=\cfrac { c }{ \lambda }

f=\cfrac { 3\times { 10 }^{ 8 } }{ 620\times { 10 }^{ -9 } }

f=4.8\times { 10 }^{ 14 }\:Hz

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