question_answer

The resistance of the wire is R ohm. Its new resistance, if it is stretched n times to its original length, will be :

A)  \[nR\]         

B)  \[\frac{n}{2}R\]

C)  \[n{{R}^{2}}\]          

D)  \[{{n}^{2}}R\]

Correct Answer: D

Solution :

Key Idea: The magnetic force acting on the particle, provides the required centripetal force to the charged particle to move on circular path. Let particle be moving with velocity v, in a magnetic field B, it follows a circular path of radius R, therefore magnetic force = centripetal force \[qvB=\frac{m{{v}^{2}}}{r}\] \[\Rightarrow \] \[v=\frac{qBr}{m}\] Energy \[E=\frac{1}{2}m{{v}^{2}}=\frac{1}{2}m{{\left( \frac{q\,Br}{m} \right)}^{2}}=\frac{{{q}^{2}}{{B}^{2}}{{r}^{2}}}{2\,m}\] Given, \[{{E}_{1}}=E,\,\,{{E}_{2}}=2\,E,\,\,{{R}_{1}}=R\] \[\therefore \] \[\frac{{{E}_{1}}}{{{E}_{2}}}={{\left( \frac{{{R}_{1}}}{{{R}_{2}}} \right)}^{2}}\] \[\Rightarrow \] \[\frac{E}{2\,E}={{\left( \frac{R}{{{R}_{2}}} \right)}^{2}}\] \[\Rightarrow \] \[\frac{1}{\sqrt{2}}=\frac{R}{{{R}_{2}}}\] \[\Rightarrow \] \[{{R}_{2}}=\sqrt{2}\,R\] Note: If particle is positively charged, it moves anticlockwise and when negatively charged it moves clockwise.