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)