- #1

goodphy

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I have several confusions regarding Faraday's law of induction.

[tex]EMF = \int_{}^{} {\vec E \cdot d\vec l} = - \frac{{d\Phi }}{{dt}} = - \frac{d}{{dt}}\int_{}^{} {\vec B \cdot d\vec S} .[/tex] It means that If the magnetic flux

*Φ*through the closed conducting loop changes in time, electric field

*E*is induced along the loop, so the current flows.

*Questions*1. Regarding the Faraday's law of induction, Lenz's law explains the induced current direction. It says the direction of induced current is such that the induced magnetic field

*B'*from this current tends to oppose the change of

*Φ*. This statement confused me to think that "secondary B-field",

*B'*, may also be included in the calculation of

*Φ*when the Faraday's law is used. But

**I think magnetic field in the**

*Φ*is the field external to the loop, not including field generated from induction process, so*B'*must be excluded. Am I right?2.

**Is Maxwell-Faraday equation**[tex]\nabla \times \vec E = - \frac{{\partial \vec B}}{{\partial t}}[/tex]

**a special case of the Faraday's law?**Stokes' theorem applying to the Maxwell-Faraday equation leads [tex]\int_{}^{} {\vec E \cdot d\vec l} = - \int\limits_{} {\frac{{\partial \vec B}}{{\partial t}}} \cdot d\vec S[/tex], which is not exactly same to the Faraday's law as shown above. It looks that It is a special case that magnetic field is time-varying while the loop is stationary (In fact, in the Maxwell-Faraday equation, a real loop is not even necessary, time-varying B-field just generates E-field no matter whether there is a real loop or not.)

3. Let's have a circular loop consisted of multiple elements, for example, metal wire + resistor, and the time-varying B-field is spatially uniform. In this case, is induced E-field uniform along the loop?

**I'm asking if there is any variation of induced E-field if the loop is made with segments of different materials.**

Please help me to clarify my mind.