A particle having mass m, charge $q$ is projected with velocity $v_{0}$ along y- axis in a region of uniform magnetic field $B_{0}$ which is outward and perpendicular to the plane of the paper as shown in the figure. The particle is continuously subjected to a frictional force which varies with velocity as ${\vec{F}}_{r} = - α \vec{V}$ , where $α$ is a constant. Consequently the particle moves on a spiral path till it comes to rest at point P. The x-co-ordinate of point P is $5 K .$ Find K?
(Take $α = 10^{- 3} k g / s; q = 10^{- 3} C; B_{0} = 1 T, v_{0} = 1 m / s, m = 20 g m$ )

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Detailed Solution

$ω = \frac{q B}{m} θ = \frac{q B}{m} t v = v_{0} e^{- \frac{α t}{m}} v_{x} = v_{0} e^{\frac{- α t}{m}} \sin \frac{q B}{m} t v_{x} = v_{0} e^{\frac{- t}{20}} \sin \frac{t}{20} x = v_{0} \int_{0}^{\infty} e^{\frac{- t}{20}} \sin \frac{t}{20} ∴ x = 10$