91影视

i) Width of conducting loop, L

ii) Resistance of the loop , R

iii) Mass of the loop, m

iv) Uniform magnetic field going into the plane of paper,$\stackrel{鈫�}{B}$

Use Faradays law of electromagnetic induction with Lenz law. The loop is moving in a uniform magnetic field so it experiences a force due to the applied magnetic field. This force must be balanced by the weight of the loop to achieve terminal velocity.

Faraday'slaw of electromagnetic inductionstates, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.

Lenz's law states that the current induced in a circuit due to a change in a magnetic field is directed to oppose the change in flux and to exert a mechanical force that opposes the motion

Formulae are as follow:

$$\begin{gathered} {\mathrm{蠒}}_{\mathrm{B}}=鈭�\stackrel{鈫�}{\mathrm{B}}.\mathrm{d}\stackrel{鈫�}{\mathrm{s}} \\ {\stackrel{,}{\mathrm{o}}}_{\mathrm{ind}}=\frac{{\mathrm{诲蠒}}_{\mathrm{B}}}{\mathrm{dt}} \\ \stackrel{鈫�}{\mathrm{F}}=\mathrm{id}\stackrel{鈫�}{\mathrm{I}}脳\stackrel{鈫�}{\mathrm{B}} \end{gathered}$$

Where,$\mathrm{桅}$is magnetic flux, B is magnetic field, i is current, 饾渶 is emf, l is length, F is force.

When the loop attains terminal velocity, its acceleration is zero. Therefore, forces acting on the loop are balanced. Therefore,

$\stackrel{鈫�}{F}=id\stackrel{鈫�}{I}脳\stackrel{鈫�}{B}=iL\left(-\hat{i}\right)脳B\left(-\hat{k}\right)=iLB\hat{j}=mg\hat{j}$

Assume y-axis to be parallel to the sides of the loop and x-axis to be parallel to the width of the loop.

$$\begin{gathered} iLB=mg \\ 鈭�i=\frac{mg}{BL} \\ \stackrel{,}{{\mathrm{o}}_{\mathrm{ind}}=-\frac{{\mathrm{诲蠒}}_{\mathrm{B}}}{\mathrm{dt}}=-\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{B}\left\{-\mathrm{dy}\right\}\mathrm{L}\right)=\mathrm{BL}.\frac{\mathrm{dy}}{\mathrm{dt}}}=BL{v}_t \end{gathered}$$

Here, dy is decreasing, so it is negative.

$$\begin{gathered} i=\frac{{{\displaystyle \stackrel{,}{o}}}_{ind}}{R}=\frac{BL{v}_t}{R}=\frac{mg}{BL} \\ \mathrm{Hence}, \\ \frac{{\mathrm{BLv}}_{\mathrm{t}}}{\mathrm{R}}=\frac{\mathrm{mg}}{\mathrm{BL}} \\ {\mathrm{v}}_{\mathrm{t}}=\frac{\mathrm{mgR}}{{\mathrm{B}}^2{\mathrm{L}}^2} \end{gathered}$$

Hence, terminal velocity of the loop is,${\mathrm{v}}_{\mathrm{t}}=\frac{\mathrm{mgR}}{{\mathrm{B}}^2{\mathrm{L}}^2}$

Therefore, Faraday鈥檚 law of electromagnetic induction and Lenz law is used to find out the emf induced in the loop.