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Question
A particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to
Options
the velocity
the momentum
the kinetic energy
none of these.
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Solution
the kinetic energy
When a particle of mass m carrying charge q is projected with speed v in a plane perpendicular to a uniform magnetic field B, the field tends to deflect the particle in a circular path of radius r.
\[\therefore \frac{m v^2}{r} = qvB\]
\[ \Rightarrow r = \frac{mv}{qB}\]
\[\text{ Now }, \]
\[\text{ Area, A } = \pi r^2 \]
\[ \Rightarrow A = \pi \left( \frac{mv}{qB} \right)^2 \]
\[ \Rightarrow A = k v^2 \]
\[\text{Here }, \]
\[k = \pi \left( \frac{m}{qB} \right)^2 \]
Kinetic energy of the particle,
Therefore, the area bounded is proportional to the kinetic energy.
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