91Ó°ÊÓ

Write the expression for the magnitude of force between two parallel straight current carrying conductors.

$F=\frac{{\mathrm{μ}}_0{i}_1{i}_2}{2\mathrm{π}d}$ …… (1)

Here,$i_1$ and $i_2$are currents through the parallel conductors,d is the distance between the two parallel conductors and ${\mathrm{μ}}_0$is the permeability.

Write the expression for the electric filed at distance of due to infinite straight conducting wire.

$E=\frac{\mathrm{λ}}{2\mathrm{Ï€}{\mathrm{Î\mu }}_{0}d}$ ……. (2)

Here,$\mathrm{λ}$ is the linear charge density and ${\mathrm{Î\mu }}_0$is the permittivity.

A line charge moving with speed constitute current that is,

$i_1=i_2=\mathrm{λ}\mathrm{Ï\dots }$

Now, let’s consider that, one of the wires, then write the expression for the attractive magnetic force on his wire per unit length.

$F=\frac{{\mathrm{μ}}_0{i}_1{i}_2}{2\mathrm{π}d}$

Substitute $\mathrm{λ}\mathrm{Ï\dots }$for $i_1$also $\mathrm{λ}\mathrm{Ï\dots }$for $i_2$.

$\begin{array}{rcl}F& =& \frac{{\mathrm{μ}}_0\left(\mathrm{λ}\mathrm{Ï\dots }\right)\left(\mathrm{λ}\mathrm{Ï\dots }\right)}{2\mathrm{Ï€}d}\\ & =& \frac{{\mathrm{μ}}_0{\mathrm{λ}}^2{\mathrm{Ï\dots }}^2}{2\mathrm{Ï€}d}\\ & & \end{array}$

Write the expression for electrostatic repulsion per unit length.

$F_C=\frac{{\mathrm{λ}}^2}{2\mathrm{Ï€}{\mathrm{Î\mu }}_{0}d}$

The line charges have to balance the magnetic attraction and electrical repulsion.

$F_C=F$ …… (3)

Substitute $\frac{{\mathrm{λ}}^2}{2\mathrm{Ï€}{\mathrm{Î\mu }}_{0}d}$for $F_C$and$\frac{{\mathrm{μ}}_0{\mathrm{λ}}^2{\mathrm{Ï\dots }}^2}{2\mathrm{Ï€}d}$for F in equation (3),

$$\begin{gathered} \frac{{\mathrm{λ}}^2}{2\mathrm{Ï€}{\mathrm{Î\mu }}_{0}d}=\frac{{\mathrm{μ}}_0{\mathrm{λ}}^2{\mathrm{Ï\dots }}^2}{2\mathrm{Ï€}d} \\ \mathrm{Ï\dots }=\frac{1}{\sqrt{{\mathrm{μ}}_0{\mathrm{Î\mu }}_0}} \end{gathered}$$

Substitute $4\mathrm{Ï€}×{10}^{-7}H/M$for${\mathrm{μ}}_0$and $8.85×{10}^{-12}{C}^2/N·m^2$for ${\mathrm{Î\mu }}_0$in above equation.

$$\begin{gathered} \mathrm{Ï\dots }=\frac{1}{\sqrt{{\mathrm{μ}}_0{\mathrm{Î\mu }}_0}} \\ \mathrm{Ï\dots }=\frac{1}{\sqrt{\left(4\mathrm{Ï€}×{10}^{-7}H/M\right)}\left(8.85×{10}^{-12}{C}^2/N.m^2\right)} \\ \mathrm{Ï\dots }=3×{10}^{8}m/s \end{gathered}$$

Thus, the required speed is $\mathrm{Ï\dots }=3×{10}^{8}m/s$and it is not reasonable two wires cannot attain that speed.