Someone asked me how Faraday's Law of Induction and Ampere's Force Law, both which form part of Maxwell's Equations, are related.(adsbygoogle = window.adsbygoogle || []).push({});

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:

[tex]F_B=Bq \times v[/tex]

[tex]B[/tex] Magnetic flux Density

[tex]q[/tex] Magnitude of charge

[tex]v[/tex] Velocity of charge

[tex]q=ALρ_q[/tex] [tex] \frac{dq}{dt}=Ap_q. \frac{dL}{dt} [/tex]

[tex]ρ_q[/tex] Charge density

[tex]A[/tex] Cross-sectional area

[tex]L[/tex] Length

[tex]v=\frac{dL}{dt}[/tex]

[tex]∴F_B=\frac{dL}{dt} \times B.ALρ_q=Ap_q. \frac{dL}{dt}×BL[/tex]

[tex]F_B=\frac{dq}{dt}×BL[/tex]

[tex]V=\frac{dW}{dq}[/tex]

[tex]V[/tex] Potential Difference

[tex]W [/tex] Work done

[tex]x[/tex] Perpendicular displacement

[tex]W=∫F_B .dx=∫\frac{dq}{dt}×BL .dx[/tex]

[tex]W=∫BL\frac{dx}{dt} .dq[/tex]

[tex]∴V=BL\frac{dx}{dt}[/tex]

[tex]BLx=\phi[/tex]

[tex]\phi[/tex] Magnetic Flux Density

Assuming B and L to be invariant:

[tex]BL\frac{dx}{dt}=\frac{d\phi}{dt}[/tex]

[tex]∴V=\frac{d\phi}{dt}[/tex]

A very crappy derivation, but it's the best possible way to show the direct connection between the two formulas.

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# Derivation of Faraday's Law from the Lorentz Force Law

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