Faraday's Law of Induction - Proof of Faraday's Law

Proof of Faraday's Law

The four Maxwell's equations (including the Maxwell–Faraday equation), along with the Lorentz force law, are a sufficient foundation to derive everything in classical electromagnetism. Therefore it is possible to "prove" Faraday's law starting with these equations. Click "show" in the box below for an outline of this proof. (In an alternative approach, not shown here but equally valid, Faraday's law could be taken as the starting point and used to "prove" the Maxwell–Faraday equation and/or other laws.)

Outline of proof of Faraday's law from Maxwell's equations and the Lorentz force law.
Consider the time-derivative of flux through a possibly moving loop, with area : The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore: where t₀ is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF (see above). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation: Next, we analyze the second term on the right-hand side: This is the most difficult part of the proof; more details and alternate approaches can be found in references. As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the time-derivative. (This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop moves with velocity v for a short time, it sweeps out a vector area vector . Therefore, the change in magnetic flux through the loop here is Therefore: where v is the velocity of a point on the loop . Putting these together, Meanwhile, EMF is defined as the energy available per unit charge that travels once around the wire loop. Therefore, by the Lorentz force law, Combining these,

Outline of proof of Faraday's law from Maxwell's equations and the Lorentz force law.

Consider the time-derivative of flux through a possibly moving loop, with area :

The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore:

where t₀ is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF (see above). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:

Next, we analyze the second term on the right-hand side:

This is the most difficult part of the proof; more details and alternate approaches can be found in references. As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the time-derivative. (This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop moves with velocity v for a short time, it sweeps out a vector area vector . Therefore, the change in magnetic flux through the loop here is

Therefore:

where v is the velocity of a point on the loop .

Putting these together,

Meanwhile, EMF is defined as the energy available per unit charge that travels once around the wire loop. Therefore, by the Lorentz force law,

Combining these,

Read more about this topic: Faraday's Law Of Induction

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