Maxwell’s Equations – Integral form of Faraday’s law (Part 5)


This is Part five of my eight part series on Maxwell Equations

I had already covered Gauss’s law for electric field in two articles and links to both these articles are

  1. Maxwell’s Equations – Integral form of Gauss’s Law (Part 1)
  2. Maxwell’s Equations – Differential form of Gauss’s Law (Part 2)

Previous two articles in this series were about Gauss’s Law for magnetic fields and their links are

Now this article would be about Integral form of Faraday’s law

Integral form of Faraday Law is given by the following equation

\(\oint\limits_C {\overrightarrow E \cdot d\overrightarrow l } = – \int\limits_s {\frac{{\partial \overrightarrow B }}{{\partial t}} \cdot } \hat nda\)
The main idea of Faraday’s Law is

Changing magnetic flux through a surface induces an emf in any boundary path of that surface, and a changing magnetic field induces a circulating electric field.

So, if flux through any surface changes, an electric field is produced along the boundary of that surface. Again if there is a conducting material present along that boundary then induced field provides an emf that produces a current through that conducting material. Moving a bar magnet through a loop of wire produces electric current in the wire but if you hold magnet stationary with respect of the loop there would be no induced current.
The negative sign in Faraday’s law tell you that the induced emf opposes the change in flux – that is, it tends to maintain the existing flux. This is called Lenz’s law.
Now I’ll try to explain the meaning of each and every symbol used in integral form of Faraday’s law by first considering the left hand side of the equation

  1. The integral on the left hand side is a line integral along a closed path C it tells us to sum the contribution from each portion of closed path \(C\) . So integral is line integral not a surface integral.
  2. \(\overrightarrow E \) is the induced electric field in V/m and is a vector quantity. This electric field is produced by the changing magnetic fields and have field lines that loop back on themselves with no points of origin or termination and have zero divergence. The electric field appearing in Faraday Law is the field measured in the reference frame of each segment \({d\overrightarrow l }\) of the path over which the circulation is calculated and this distinction is made only because it is only in this frame that the electric field lines actually circulate back to themselves as can be seen in the figure given belowfield
  3. Dot product tells you to find the part of \(\overrightarrow E \) parallel to \({d\overrightarrow l }\) (along path C)
  4. \({d\overrightarrow l }\) is an incremental segment of path  \(C\)
  5. Right side as a whole is the flux of the time rate of the change of magnetic field.
  6. \({\frac{{\partial \overrightarrow B }}{{\partial t}}}\) is the rate of change of magnetic field with time.

We can use Faraday’s law and the flux rule in solving a variety of problems that have changing magnetic flux and induced electric fields, in particular problems of two types:
(1) Given information about the changing magnetic flux, find the induced emf.
(2) Given the induced emf on a specified path, determine the rate of change of the magnetic field magnitude or direction or the area bounded by the path.
In situations of high symmetry, in addition to finding the induced emf, it is also possible to find the induced electric field when the rate of change of the magnetic field is known.

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