- #1
iScience
- 466
- 5
$$\vec{F}=q\vec{v}\times\vec{B}$$
$$\frac{d\vec{F}}{dq}=\vec{v}\times\vec{B}$$
$$\int\frac{d\vec{F}}{dq} \cdot ds=\int(\frac{d\vec{s}}{dt}\times\vec{B}) \cdot ds$$
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/dtThat 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 a lot further with the second "method," but is this a valid derivation? and what went wrong with the first method?
$$\frac{d\vec{F}}{dq}=\vec{v}\times\vec{B}$$
$$\int\frac{d\vec{F}}{dq} \cdot ds=\int(\frac{d\vec{s}}{dt}\times\vec{B}) \cdot ds$$
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/dtThat 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 a lot further with the second "method," but is this a valid derivation? and what went wrong with the first method?
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