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

## Main Question or Discussion Point

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 Density

Assuming 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.

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Hope I haven't made any mistakes.

BruceW
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