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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.
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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