Derivation of Faraday's Law from the Lorentz Force Law

[Lamarr]

Someone asked me how Faraday's Law of Induction and Ampere's Force Law, both which form part of Maxwell's Equations, are related.

Ampere's Force Law is derived from the Lorentz Force Law. They are entirely compatible with Faraday's Law of Induction. Here's how...The Lorentz Force Law states:

$$F_B=Bq \times v$$

$$B$$ Magnetic flux Density

$$q$$ Magnitude of charge

$$v$$ Velocity of charge

$$q=ALρ_q$$ $$\frac{dq}{dt}=Ap_q. \frac{dL}{dt}$$

$$ρ_q$$ Charge density

$$A$$ Cross-sectional area

$$L$$ Length
$$v=\frac{dL}{dt}$$
$$∴F_B=\frac{dL}{dt} \times B.ALρ_q=Ap_q. \frac{dL}{dt}×BL$$
$$F_B=\frac{dq}{dt}×BL$$$$V=\frac{dW}{dq}$$

$$V$$ Potential Difference

$$W$$ Work done

$$x$$ Perpendicular displacement$$W=∫F_B .dx=∫\frac{dq}{dt}×BL .dx$$

$$W=∫BL\frac{dx}{dt} .dq$$

$$∴V=BL\frac{dx}{dt}$$
$$BLx=\phi$$

$$\phi$$ Magnetic Flux DensityAssuming B and L to be invariant:

$$BL\frac{dx}{dt}=\frac{d\phi}{dt}$$

$$∴V=\frac{d\phi}{dt}$$
A very crappy derivation, but it's the best possible way to show the direct connection between the two formulas.

 

[Lamarr]

Hope I haven't made any mistakes.

 

[BruceW]

This page on wikipedia also has a derivation of this: http://en.wikipedia.org/wiki/Faraday's_law_of_induction

And it has references 22,23,24 that might give details on the derivation. I am interested in this myself, so I'll probably check them out too :)

 

[andrien]

Are you familiar with the fact that the emf induced comes from two parts one is the motion part which you are counting and the other is flux change(both are different).think about faraday disk and see how will you apply flux rule to count for emf.

 

[Lamarr]

Well flux change is due to motion as well.