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  • A circular wire loop of radius r can withstand a radial force T before breaking. A particle of mass m and charge q(q > 0) is sliding over the wire. A magnetic field B is applied normal to the pl

A circular wire loop of radius r can withstand a radial force T before breaking. A particle of mass m and charge q(q > 0) is sliding over the wire. A magnetic field B is applied normal to the plane of the wire. The maximum speed \mathrm{V}_{\max } the particle can have before the loop breaks is

Option: 1

Zero


Option: 2

\mathrm{\sqrt{\frac{r T}{m}}}


Option: 3

\mathrm{\frac{r}{2}\left[q B+\sqrt{q^2 B^2+\frac{4 T}{m r}}\right]}


Option: 4

\mathrm{\frac{r}{2}\left[\frac{q B}{m}+\sqrt{\frac{q^2 B^2}{m^2}+\frac{4 T}{m r}}\right]}


Answers (1)

best_answer

The following forces act on the particle.

• Force T acting radially inwards.

• Centrifugal force \mathrm{\frac{m v^2}{r}} acting radially outwards

• Magnetic force qvB acting radially inwards.

\mathrm{\begin{array}{ll} \Rightarrow & T+q v B=\frac{m v^2}{r} \\ \Rightarrow & \frac{m v^2}{r}-q v B-T=0 \\ \Rightarrow & v^2=\left(\frac{q B r}{m}\right) v-\frac{T r}{m}=0 \\ \Rightarrow & v=\frac{1}{2}\left[\frac{q B r}{m}+\sqrt{\frac{q^2 B^2 r^2}{m^2}+\frac{4 T r}{m}}\right] \\ \Rightarrow \quad & v=\frac{r}{2}\left[\frac{q B}{m}+\sqrt{\frac{q^2 m^2}{m^2}+\frac{4 T}{m r}}\right] \end{array}}

Posted by

Devendra Khairwa

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