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

विकल्प

• the velocity

• the momentum

• the kinetic energy

• none of these.

Answer 1

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,

\[E = \frac{1}{2}m v^2\]

Therefore, the area bounded is proportional to the kinetic energy.