question_answer

Two infinitely long parallel wires having linear charge densities \[{{\lambda }_{1}}\] and \[{{\lambda }_{2}}\] respectively are placed at a distance of R metres. The force per unit length on either wire will be \[\left( K=\frac{1}{4\pi {{\varepsilon }_{0}}} \right)\]   [MP PMT/PET 1998; DPMT 2000]

A)                    \[K\frac{2{{\lambda }_{1}}{{\lambda }_{2}}}{{{R}^{2}}}\]

B)                                      \[K\frac{2{{\lambda }_{1}}{{\lambda }_{2}}}{R}\]

C)                    \[K\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{{{R}^{2}}}\]              

D)            \[K\frac{{{\lambda }_{1}}{{\lambda }_{2}}}{R}\]

Correct Answer: B

Solution :

           Force on l length of the wire 2 is \[{{F}_{2}}=Q{{E}_{1}}=({{\lambda }_{2}}l)\frac{2k{{\lambda }_{1}}}{R}\] Þ \[\frac{{{F}_{2}}}{l}=\frac{2k{{\lambda }_{1}}{{\lambda }_{2}}}{R}\] Also \[\frac{{{F}_{1}}}{l}=\frac{{{F}_{2}}}{l}=\frac{F}{l}=\frac{2k{{\lambda }_{1}}{{\lambda }_{2}}}{R}\]