Consider a cylindrical region of uniform but time varying magnetic field. Now consider a plane perpendicular to this field. Top view of this cylindrical magnetic field on the plane is shown in the diagram which is a circular region of radius $r_{0} = 1 m$ . A particle of mass $π k g$ and charge 2C is projected with velocity $v_{0} = \sqrt{5} m / s$ , tangential to the circle from point A. This particle intersects another radial line OB at point ‘B’ with velocity ‘v’. If the magnetic field increasing at the rate of 4T/s then find the value of ‘v’. (in m/s) $(∠ A O B = 90^{0})$ .

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answer is 3.

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Detailed Solution

Around OABO
$\oint \vec{E} \cdot d \vec{r} = - \frac{d ϕ}{d t}$
$\int_{A}^{B} \vec{E} \cdot d \vec{r} = \frac{d}{d t} [\frac{B π R^{2}}{4}]$ -- (1)
But $\int_{A}^{B} Q \vec{E} \cdot d \vec{r} = \frac{1}{2} m V_{B}^{2} - \frac{1}{2} m V_{A}^{2}$ -- (2)
From (1) & (2) $V = 3 m / s$ .