A particle of charge ‘?q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘’. Then time period will be given as

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Q.
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

1 $T^{2} = \frac{4 π^{2} m}{2 k λ q} r^{3}$

2 $T = 2 π r \sqrt{\frac{m}{2 k λ q}}$

3 $T = \frac{1}{2 π r} \sqrt{\frac{m}{2 k λ q}}$

4 $T = \frac{1}{2 π} \sqrt{\frac{2 k λ q}{m}}$

Ans.
(2)

$F_{e} = \frac{m v^{2}}{r} \Rightarrow q E = \frac{m v^{2}}{r}$

$q (\frac{2 K λ}{r}) = \frac{m v^{2}}{r} \Rightarrow v = \sqrt{\frac{2 K q λ}{m}}$

$T = \frac{2 π r}{v} = \frac{2 π r}{\sqrt{\frac{2 K q λ}{m}}} = 2 π r \sqrt{\frac{m}{2 K q λ}}$

Solution

Q.
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

1 $T^{2} = \frac{4 π^{2} m}{2 k λ q} r^{3}$

2 $T = 2 π r \sqrt{\frac{m}{2 k λ q}}$

3 $T = \frac{1}{2 π r} \sqrt{\frac{m}{2 k λ q}}$

4 $T = \frac{1}{2 π} \sqrt{\frac{2 k λ q}{m}}$

Ans.
(2)

$F_{e} = \frac{m v^{2}}{r} \Rightarrow q E = \frac{m v^{2}}{r}$

$q (\frac{2 K λ}{r}) = \frac{m v^{2}}{r} \Rightarrow v = \sqrt{\frac{2 K q λ}{m}}$

$T = \frac{2 π r}{v} = \frac{2 π r}{\sqrt{\frac{2 K q λ}{m}}} = 2 π r \sqrt{\frac{m}{2 K q λ}}$

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Solution

Q. A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘+λ’. Then time period will be given as (Consider k as Coulomb’s constant) [2024]

1 T2=4π2m2kλqr3

2 T=2πrm2kλq

3 T=12πrm2kλq

4 T=12π2kλqm

Ans. (2) Fe=mv2r⇒qE=mv2r q(2Kλr)=mv2r⇒v=2Kqλm T=2πrv=2πr2Kqλm=2πrm2Kqλ

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Q.
A particle of charge ‘−q’ and mass ‘m’ moves in a circle of radius ‘r’ around an infinitely long line charge of linear density ‘ $+ λ$ ’. Then time period will be given as

(Consider k as Coulomb’s constant) [2024]

1 $T^{2} = \frac{4 π^{2} m}{2 k λ q} r^{3}$

2 $T = 2 π r \sqrt{\frac{m}{2 k λ q}}$

3 $T = \frac{1}{2 π r} \sqrt{\frac{m}{2 k λ q}}$

4 $T = \frac{1}{2 π} \sqrt{\frac{2 k λ q}{m}}$

Ans.
(2)

$F_{e} = \frac{m v^{2}}{r} \Rightarrow q E = \frac{m v^{2}}{r}$

$q (\frac{2 K λ}{r}) = \frac{m v^{2}}{r} \Rightarrow v = \sqrt{\frac{2 K q λ}{m}}$

$T = \frac{2 π r}{v} = \frac{2 π r}{\sqrt{\frac{2 K q λ}{m}}} = 2 π r \sqrt{\frac{m}{2 K q λ}}$