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 \[\left( k=\frac{1}{4\pi {{\varepsilon }_{0}}} \right)\]

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 :

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