91Ó°ÊÓ

The force per unit length between two current carrying wires is given by

$\frac{\mathrm{F}}{\mathrm{L}}=\frac{{\mathrm{μ}}_0{âˆ\yen }_1}{2\mathrm{Ï€}d}$

Where, ${\mathrm{μ}}_0$ is the permeability of vaccum,$l_1$ and ${\mathit{l}}_2$is the current through the two wires and d is the distance between wire.

$I=\frac{I_0}{\sqrt{2}}$

RMS Value of current

It is the value of alternating current which is given by direct current which flows through a resistance.

Q = CV

Mathematically it is given by

$\frac{\mathrm{F}}{\mathrm{L}}=\frac{{\left.{\mathrm{μ}}_0\right|}^2}{2\mathrm{π}d}$

Charge on the capacitor

The charge on the capacitor is given by

Q = CV

The same current l is flowing through the wires

$\frac{\mathrm{F}}{\mathrm{L}}=\frac{{\left.{\mathrm{μ}}_0\right|}^2}{2\mathrm{π}d}$

From Newton’s second law

localid="1668271320115" $\frac{ma}{\mathrm{L}}=\frac{{\left.{\mathrm{μ}}_0\right|}^2}{2\mathrm{π}d}$

_{Now .put the value RMS current}

_{$\frac{ma}{\mathrm{L}}=\frac{{\mathrm{μ}}_0}{2\mathrm{π}d}{\left(\frac{I_0}{\sqrt{2}}\right)}^2$}

_{From ohm’s law}

_{$\frac{ma}{\mathrm{L}}=\frac{{\mathrm{μ}}_0}{4\mathrm{π}d}{\left(\frac{I_0}{\sqrt{2}}\right)}^2$}

In terms of charge on the capacitor

$\begin{array}{r}\mathrm{λ}a=\frac{{\mathrm{μ}}_0}{4\mathrm{π}d}{\left(\frac{Q_0}{RC}\right)}^2\\ a=\frac{{\mathrm{μ}}_0{Q}_0^2}{4\mathrm{π}\mathrm{λ}{R}^2{C}^{2}d}\end{array}$

Thus, acceleration of wire is$a=\frac{{\mathrm{μ}}_0{Q}_0^2}{4\mathrm{π}\mathrm{λ}{R}^2{C}^{2}d}$

The velocity of wire after time equals to time constant

$\begin{array}{r}v=at\\ v=\frac{{\mathrm{μ}}_0{Q}_0^{2}RC}{4\mathrm{π}\mathrm{λ}{R}^2{C}^{2}d}\\ v=\frac{{\mathrm{μ}}_0{Q}_0^2}{4\mathrm{π}RRCd}\end{array}$

According to energy conservation

$\begin{array}{r}\frac{1}{2}m{v}^2=mgh\\ h=\frac{1}{2}\frac{v^2}{g}\end{array}$

Now putting the values of constants in above expression

$h=\frac{1}{2g}{\left(\frac{{\mathrm{μ}}_0{Q}_0^2}{4\mathrm{π}\mathrm{λ}RCd}\right)}^2$

Thus, the wireswill rise to height$h=\frac{1}{2g}{\left(\frac{{\mathrm{μ}}_0{Q}_0^2}{4\mathrm{π}\mathrm{λ}RCd}\right)}^2$