- #1
goodphy
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Hello.
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.
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.