Consider the situation of the previous problem. 1. Calculate the … - Vidyadip

Question

Consider the situation of the previous problem.

Calculate the force needed to keep the sliding wire moving with a constant velocity $v$.
If the force needed just after $t = 0$ is $F_0$, find the time at which the force needed will be $\frac{\text{F}_0}{2}.$

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Answer

$\text{e}=\text{Bvl}$
$\text{i}=\frac{\text{e}}{\text{R}}=\frac{\text{Bvl}}{2\text{r}(\text{l}+\text{vt})}$

$\text{F}=\text{ilB}=\frac{\text{Bvl}}{2\text{r}(\text{l}+\text{vt})}\times\text{lB}=\frac{\text{B}^2\text{l}^2\text{v}}{2\text{r}(\text{l}+\text{vt})}$
Just after $\text{t}=0$

$\text{f}_0=\text{ilB}=\text{lB}\Big(\frac{\text{lBv}}{2\text{rl}}\Big)=\frac{\text{l}\text{B}^2\text{v}}{2\text{r}}$
$\frac{\text{f}_0}{2}=\frac{\text{l}\text{B}^2\text{v}}{4\text{r}}=\frac{\text{l}^2\text{B}^2\text{v}}{2\text{r}(\text{l}+\text{vt})}$
$\Rightarrow2\text{l}=\text{l}+\text{vt}$
$\Rightarrow\text{T}=\frac{\text{l}}{\text{v}}$

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