Faraday's law of Electromagnetic Induction ~ black-sci

Michael Faraday, who became one of the greatest scientists of the 19th century, began his career as a chemist. He worked as an assistant to famous chemist Humphry Davy. Faraday had no formal education beyond basic reading, writing and math, and he never went to college.

At the age of 14, he became an apprentice to a local bookbinder and bookseller. During his seven-year apprenticeship, Faraday read many books and developed interest in science. His main discoveries include the principles underlying electromagnetic induction, diamagnetism and electrolysis and the invention of electric motor and dynamo.

Faraday's Law states that the magnitude of emf induced in the coil is equal to the rate of change of flux that linkages with the coil. The flux linkage of the coil is the product of the number of turns in the coil and flux associated with the coil.

Faraday's law can be understood in-depth from Maxwell's equation

$e= \oint E.dl =-\frac{d\phi_{B} }{dt}$

The above equation means If there is time-varying magnetic flux passing through any arbitrary loop drawn in space, then there is an electric field established such that the line integral of electric field E on a closed loop is equal to the rate of change of flux.

Note here that it is not required for the conductor or coil to be present to establish the electric field and hence emf. But generally, no current flows due to the high resistance of air.

Let's say we place a circular copper coil in a time-varying magnetic field, then free electrons in the copper will start moving in the direction of the electric field created as stated above. The current flowing, in turn, produces its own magnetic field and tries to effectively cancel out variations in original magnetic field. This is in accordance with Lenz's Law.

If $\phi=\phi _{m} sin(\omega t)$

$e=-\frac{d\phi }{dt}=-\omega \phi _{m}cos(\omega t)$ ...............................................(1)

$e= \oint E.dl =E.2\pi r$ .......................................................(2)

Equating 1 and 2, we get

$E=-\frac{\omega \phi _{m}cos(\omega t)}{2\pi r}$

Suppose one turn coil replaced with N turn coil, then induced emf is simply N times that of single turn emf. This is because for each turn $e_{1}=-\frac{d\phi }{dt}$ and so for N turns in series ,the emf will add up.

$e=Ne_{1}=N\frac{d\phi }{dt}$

Here N being multiplied with the rate of change of flux is a very critical concept in Faraday's law. It is essential that to apply Maxwell's equation loop must be closed since phi is flux passing through a closed loop. A turn is almost circle, but it is not a closed-loop, though, as said before, it is not required the conductor to be present; hence we take an arbitrary circle in space that has the same radius as the turn and then electric field E obtained is same as single turn coil. It is as if the circles of the electric field are already present in space, and we are just placing a coil there.

$e_{1}=e_{2}=...=e_{n}=\oint E.dl=-\frac{d\phi }{dt}$

$E=-\frac{\omega \phi _{m}cos(\omega t)}{2\pi r}$

$e=e_{1}+e_{2}+...+e_{n}= -N\frac{d\phi }{dt}$

Since N is a constant

$e=-\frac{d(N\phi )}{dt}=-\frac{d\psi }{dt}$

$\psi=N\phi$

Psi is called flux linkage, and instead of adding emf of all turns in series, it is dealt with by taking emf induced equal to rate change of flux linkage.

Faraday's law explains the working of electrical elements like Inductor and Transformer and is the basis for converting mechanical motion into electrical energy in all electric generators.

black-sci

Menu

Wednesday, May 18, 2022

Faraday's law of Electromagnetic Induction

0 comments:

Post a Comment

Blog Archive

Categories