A particle of charge q and mass m is projected with a velocity V0 towards a circular region having uniform magnetic field B perpendicular and into the plane of paper from point P as shown in the figure. R is the radius and O is the centre of the circular region. If the line OP makes angle θ with the direction of V0 , then the value of V0 so that particle passes through O is

Question

A particle of charge q and mass m is projected with a velocity V0 towards a circular region having uniform magnetic field B perpendicular and into the plane of paper from point P as shown in the figure. R is the radius and O is the centre of the circular region. If the line OP makes angle θ with the direction of V0 , then the value of V0 so that particle passes through O is <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-ptxuezxmdkui.png" /> (A) (q B R/m sin θ ) (B) (q B R/2 m sin θ ) (C) (2 q B R/m sin θ ) (D) (3 q B R/2 m sin θ )

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-ptxuezxm2lbi.png" /> r=(m V0/q B) R=2r sinθ ⇒ R=(2 m V0/q B)sinθ ⇒ V0=(q B R/2 m sin θ )

$\frac{qBR}{m s in θ}$

$\frac{qBR}{2 m s in θ}$

$\frac{2 qBR}{m s in θ}$

$\frac{3 qBR}{2 m s in θ}$

$r = \frac{m V _{0}}{qB}$
$R = 2 r s in θ$
$\Rightarrow R = \frac{2 m V _{0}}{qB} s in θ$
$\Rightarrow V_{0} = \frac{qBR}{2 m s in θ}$

msinθqBR​

2msinθqBR​

msinθ2qBR​

2msinθ3qBR​

r=qBmV0​​ R=2rsinθ ⇒R=qB2mV0​​sinθ ⇒V0​=2msinθqBR​

$\frac{qBR}{m s in θ}$

$\frac{qBR}{2 m s in θ}$

$\frac{2 qBR}{m s in θ}$

$\frac{3 qBR}{2 m s in θ}$

$r = \frac{m V _{0}}{qB}$
$R = 2 r s in θ$
$\Rightarrow R = \frac{2 m V _{0}}{qB} s in θ$
$\Rightarrow V_{0} = \frac{qBR}{2 m s in θ}$