91影视

The length of the loop is $l$.

The resistance of the loop is $R$.

The self-inductance is $L$.

The mass of the loop is $m$.

The generated current in the loop is $I=\frac{vBh}{R}$.

The expression to calculate the back emf is given as follows.

$蔚=Bhv$ 鈥︹��. (1)

The expression to calculate the back emf associated with the self-inductance is given as follows.

$蔚=-L\frac{dI}{dt}$ 鈥︹��. (2)

The expression for the force on the wire is given as follows.

$F=Blh$ 鈥︹��.. (3)

The expression for the force on the wire in terms of mass is given as follows.

$F=m\frac{dv}{dt}$ 鈥︹��.. (4)

Equate the forces on the wire.

From the equation (3) and (4).

$\begin{array}{c}m\frac{dv}{dt}=Blh\\ \frac{dv}{dt}=\frac{Blh}{m}\end{array}$

Differentiate the above equation with respect to $t$.

$\frac{d^{2}v}{dt}=\frac{Bl}{m}\frac{dI}{dt}$ 鈥︹�︹�� (5)

Equate the equation (1) and (2).

$\begin{array}{l}鈭�L\frac{dI}{dt}=Bhv\\ \frac{dI}{dt}=鈭�\frac{Bhv}{L}\end{array}$

Substitute $鈭�\frac{Bhv}{L}$for $\frac{dI}{dt}$into equation (5).

$\begin{array}{l}\frac{d^{2}v}{dt}=\frac{Bl}{m}(鈭�\frac{Bhv}{L})\\ \frac{d^{2}v}{dt}=鈭�\frac{B^{2}lhv}{mL}\end{array}$

From the above equation,

$\begin{array}{l}{蝇}^2=\frac{B^2{h}^2}{mL}\\ 蝇=\sqrt{\frac{B^2{h}^2}{mL}}\\ 蝇=\frac{Bh}{\sqrt{mL}}\end{array}$

Hence the expression for the frequency is $\frac{Bh}{\sqrt{mL}}$.