A sliding conductor

Electricity and Magnetism Level 4

A horizontal conducting rod of mass \(m\) and length \(L\) is free to slide on two vertical conducting rails as shown, where there is a magnetised field of intensity \(B\) directed inside the screen and the resistance of the resistor is \(R\). Find the distance travelled by the rod after \(t\) seconds.

Inspiration.

\[d=\dfrac{mgR}{B^2L^2}\left(t-\dfrac{mR}{B^2L^2}\left(1+e^{-\frac{B^2L^2t}{mR}}\right)\right)\] \[d=\dfrac{mgR}{B^2L^2}\left(t-\dfrac{mR}{B^2L^2}\left(1-e^{-\frac{B^2L^2t}{mR}}\right)\right)\] \[d=\dfrac{mgR}{B^2L^2}\left(t+\dfrac{mR}{B^2L^2}e^{-\frac{B^2L^2t}{mR}}\right)\] \[d=\dfrac{mgR}{B^2L^2}\left(t+\dfrac{mR}{B^2L^2}\left(1+e^{-\frac{B^2L^2t}{mR}}\right)\right)\] \[d=\dfrac{mgR}{B^2L^2}\left(t+\dfrac{mR}{B^2L^2}\left(1-e^{-\frac{B^2L^2t}{mR}}\right)\right)\] \[d=\dfrac{mgR}{B^2L^2}\left(t-\dfrac{mR}{B^2L^2}e^{-\frac{B^2L^2t}{mR}}\right)\]

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