An overview of Faraday’s Law

Walter F. Smith, Haverford College 6-5-02


Text Box:  Michael Faraday is now generally recognized as the greatest experimental physicist in history, but his beginnings were humble.  His father was a blacksmith working near London.  Faraday received no formal education, but spent several years working as errand boy and apprentice to two bookbinders, and apparently developed an interest in science through the books he worked with.  At the age of 21, a life-changing event occurred.  As described by the BBC history of him, “…after being given tickets by a satisfied customer, Faraday attended the last 4 lectures given by the chemist, Humphry Davy [pioneer in electrochemistry and discoverer of sodium and potassium], at the Royal Institution. At each lecture Faraday took copious amounts of notes, which he later wrote up, bound and presented to Davy. At the same time, he also applied for a job … On the 24th December 1812, Davy’s carriage rolled up outside Faraday's home in Weymouth Street, London. A surprised Faraday was given a note asking him to appear at the Royal Institution, which he did. Later, he was taken on as Davy's assistant ...” 

Text Box:  Faraday made critical discoveries in electrochemistry, discovered electromagnetic induction, and invented the electric motor and the electric generator.  Because he was not mathematically inclined, he may have found it difficult to think about electricity and magnetism in the highly abstract way used by some of his contemporaries (e.g. Gauss).  So, he invented the much more visual concepts of the electric field E and the magnetic field B, which now form the entire basis for our understanding of electricity and magnetism.  He was a man of strong faith, and refused to help in the development of tools of war, such as poison gas.  Faraday’s name is spread throughout science:  Farad (the unit used to quantify capacitors), Faradaic current (electrical current carried by ions in a solution), and of course Faraday’s Law:

The left side describes the amount of variation in space of the electric field E, and is called the “circulation of E”, where E can be pronounced “vector cap E”,  since the electric field is a vector (i.e. it has a direction as well as a magnitude).  The right side describes the amount of variation in time of the magnetic field B:  The  part is called the “magnetic flux”, and describes the number of magnetic field lines that pierce a surface; it can be written more compactly as , pronounced “feye sub B”.  The part (pronounced “d by d t”) means “rate of change”.  Putting this all together, Faraday’s law says that a magnetic field that changes in time can create an electric field!  This realization forms the basis for electric generators and transformers. (Transformers are used to increase or decrease the voltage in a circuit; this ability is absolutely essential for transmission of electric power.)

Later, Maxwell used Faraday’s ideas of electric and magnetic fields to synthesize the four “Maxwell’s equations”, of which Faraday’s law is one.  These equations give the relationships between the electric field, the magnetic field, and charge.  Of the four, Faraday’s law is the most Text Box:  
Michael Faraday
beloved of students and professors, partly because it has so many important technological and scientific implications, but perhaps also because it’s easy to make up homework problems about it, and students seem to enjoy solving them!

Faraday’s law is very similar in appearance to the corrected version of Ampère’s law which was deduced by Maxwell:

Faraday’s law:

Ampère’s law: 

Text Box:  
James Clerk Maxwell, his wife Katherine, and their dog.  Maxwell was well known for talking to his dog about his scientific theories.
The main differences are the exchange of the roles of E and B, the addition of a few constants, and the term, which represents the flow of electric charges or “monopoles”.  Since there are no magnetic monopoles, there is no equivalent term in Faraday’s law. 

Maxwell saw that the combination of these two laws explained the nature of electromagnetic radiation (light, radio waves, x-rays, etc.): Faraday’s law says that a changing B creates an E.  Ampère’s law says that a changing E creates a B.  Maxwell showed that this meant a wave could propagate, in which the changing B created a changing E, which in turn created a changing B, and so on, so that the wave is self-sustaining.  However, he showed that this all worked only if the wave propagates at a particular speed: 186,283 miles per second, i.e. the speed of light.  The fact that Maxwell’s derivation of this speed was based on basic physical laws led Einstein to assume that the speed of light must be the same in all reference frames, which in turn led to his theory of relativity.