Question

• State - Faraday's law of induction.

Answer 1

Faraday's Law. Now that we have a basic understanding of the magnetic field, we are ready to define Faraday's Law of Induction. It states that the induced voltage in a circuit is proportional to the rate of change over time of the magnetic flux through that circuit.

Answer 2

{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma (t)}\mathbf {B} (t)\cdot \mathrm {d} \mathbf {A} }{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma (t)}\mathbf {B} (t)\cdot \mathrm {d} \mathbf {A} }

(by definition). This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation and some vector identities; the details are in the box below:

{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}=-\oint _{\partial \Sigma }\left(\mathbf {E} +\mathbf {v} _{\mathbf {l} }\times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} .}{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}=-\oint _{\partial \Sigma }\left(\mathbf {E} +\mathbf {v} _{\mathbf {l} }\times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} .}

where ∂Σ is the boundary (loop) of the surface Σ, and vl is the velocity of a part of the boundary.

In the case of a conductive loop, EMF (Electromotive Force) is the electromagnetic work done on a unit charge when it has traveled around the loop once, and this work is done by the Lorentz force. Therefore, EMF is expressed as

{\displaystyle {\mathcal {E}}=\oint \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} }{\displaystyle {\mathcal {E}}=\oint \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} }

where {\displaystyle {\mathcal {E}}}{\mathcal {E}} is EMF and v is the unit charge velocity.

In a macroscopic view, for charges on a segment of the loop, v consists of two components in average; one is the velocity of the charge along the segment vt, and the other is the velocity of the segment vl (the loop is deformed or moved). vt does not contribute to the work done on the charge since the direction of vt is same to the direction of {\displaystyle d\mathbf {l} }d\mathbf {l} . Mathematically,

{\displaystyle (\mathbf {v} \times B)\cdot d\mathbf {l} =((\mathbf {v} _{t}+\mathbf {v} _{l})\times B)\cdot d\mathbf {l} =(\mathbf {v} _{t}\times B+\mathbf {v} _{l}\times B)\cdot d\mathbf {l} =(\mathbf {v} _{l}\times B)\cdot d\mathbf {l} }{\displaystyle (\mathbf {v} \times B)\cdot d\mathbf {l} =((\mathbf {v} _{t}+\mathbf {v} _{l})\times B)\cdot d\mathbf {l} =(\mathbf {v} _{t}\times B+\mathbf {v} _{l}\times B)\cdot d\mathbf {l} =(\mathbf {v} _{l}\times B)\cdot d\mathbf {l} }

since {\displaystyle (\mathbf {v} _{t}\times B)}{\displaystyle (\mathbf {v} _{t}\times B)} is perpendicular to {\displaystyle d\mathbf {l} }d\mathbf {l}  as {\displaystyle \mathbf {v} _{t}}{\displaystyle \mathbf {v} _{t}} and {\displaystyle d\mathbf {l} }d\mathbf {l}  are along the same direction. Now we can see that, for the conductive loop, EMF is same to the time-derivative of the magnetic flux through the loop except for the sign on it. Therefore, we now reach the equation of Faraday's law (for the conductive loop) as

{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}=-{\mathcal {E}}}{\displaystyle {\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}=-{\mathcal {E}}}

where {\displaystyle {\mathcal {E}}=\oint \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} }{\displaystyle {\mathcal {E}}=\oint \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} }. With breaking this integral, {\displaystyle \oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} }{\displaystyle \oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} } is for the transformer EMF (due to a time-varying magnetic field) and {\displaystyle \oint \left(\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} =\oint \left(\mathbf {v} _{l}\times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} }{\displaystyle \oint \left(\mathbf {v} \times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} =\oint \left(\mathbf {v} _{l}\times \mathbf {B} \right)\cdot \mathrm {d} \mathbf {l} } is for the motional EMF (due to the magnetic Lorentz force on charges by the motion or deformation of the loop in the magnetic field).

EMF for non-thin-wire circuits

It is tempting to generalize Faraday's law to state: If ∂Σ is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through Σ equals the EMF around ∂Σ. This statement, however, is not always true and the reason is not just from the obvious reason that EMF is undefined in empty space when no conductor is present. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve ∂Σ matches the actual velocity of the material conducting the electricity. The two examples illustrated below show that one often obtains incorrect results when the motion of ∂Σ is divorced from the motion of the material.

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