- #1
masteralien
- 36
- 2
- TL;DR
- There seems to be a contradiction in the sign of the motional EMF for a spinning disk depending in the formula used
The formula for motional EMF is
$$\oint({\bf{v}}\times{\bf{B}})d{\bf{l}}=-\frac{d}{dt}\int{{\bf{B}}\cdot{\bf{\hat{n}}}da}$$However applying this for a rotating disk of radius a there seems to be a sign contradiction
$${\bf{v}}\times{\bf{B}}=\omega s{\bf{\hat{\varphi}}}\times B{\bf{\hat{z}}}=B\omega s {\bf{\hat{s}}}$$
$$\int^{a}_0{B\omega s}ds=\frac{1}{2}B\omega a^2$$Now doing it with the Double Integral by moving the derivative inside
$$
-\frac{d}{dt}\int^{2\pi}_0\int^{a}_0{Bsdsd\varphi}$$
$$\\\frac{d\varphi}{dt}=\omega$$
$$\\-\int^{a}_0{B\omega sds}=-\frac{1}{2}B\omega a^2$$
These expressions are similar but have the opposite sign why is this.
My question is why is there this contradiction here did I do something wrong like these formulas should be the same.
$$\oint({\bf{v}}\times{\bf{B}})d{\bf{l}}=-\frac{d}{dt}\int{{\bf{B}}\cdot{\bf{\hat{n}}}da}$$However applying this for a rotating disk of radius a there seems to be a sign contradiction
$${\bf{v}}\times{\bf{B}}=\omega s{\bf{\hat{\varphi}}}\times B{\bf{\hat{z}}}=B\omega s {\bf{\hat{s}}}$$
$$\int^{a}_0{B\omega s}ds=\frac{1}{2}B\omega a^2$$Now doing it with the Double Integral by moving the derivative inside
$$
-\frac{d}{dt}\int^{2\pi}_0\int^{a}_0{Bsdsd\varphi}$$
$$\\\frac{d\varphi}{dt}=\omega$$
$$\\-\int^{a}_0{B\omega sds}=-\frac{1}{2}B\omega a^2$$
These expressions are similar but have the opposite sign why is this.
My question is why is there this contradiction here did I do something wrong like these formulas should be the same.
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