# Derivation of Faraday's Law from Lorentz's

1. Aug 7, 2014

### iScience

$$\vec{F}=q\vec{v}\times\vec{B}$$

$$\frac{d\vec{F}}{dq}=\vec{v}\times\vec{B}$$

$$\int\frac{d\vec{F}}{dq} \cdot ds=\int(\frac{d\vec{s}}{dt}\times\vec{B}) \cdot ds$$

from here, I went about it two different ways:

1.) Here I assumed everything was at right angles and got rid of all the vectors and vector products

$$\varepsilon=\int \frac{ds}{dt}B ds=\int \frac{ds}{dt}B \frac{ds}{dt}dt$$

By u substitution

$$u=\frac{ds}{dt}, du=dt$$
$$\varepsilon=\int B(u^2)du=\frac{Bv^3}{3}$$

where v = ds/dt

That was the first way i went about it, but i didn't feel any closer to Faraday's law.

2.) Here I left the vectors alone on the RHS; I figured since $\hat{v}$ and d$\hat{s}$ were perpendicular, the quantity ($\vec{v}$s) would be a time derivative of the area formed

$$\varepsilon=\int\frac{ds}{dt}B ds=\int(\vec{v}\times\vec{B}) \cdot d\vec{s}=\dot{A}B$$

$$\varepsilon=\frac{BA}{dt}$$

don't know where the minus sign is; probably was supposed to do something with the cross product, but didn't know what.

Well I got alot further with the second "method," but is this a valid derivation? and what went wrong with the first method?

Last edited: Aug 7, 2014
2. Aug 9, 2014

### iScience

Oh! PS, well, more like pre-script... my goal is to derive faraday's induction law from lorentz force law