Correct Answer - Option 1 : inversely proportional to its mass
The correct answer is option 1) i.e. inversely proportional to its mass
CONCEPT:
- A cyclotron is a particle accelerator that accelerates charged particles to higher energies.
- The working principle of a cyclotron is the Lorentz force.
- The charged particle is placed at the center of the arrangement between the poles of two electromagnets.
- The magnetic field causes the particle to trace a circular path as it experiences a Lorentz force.
- An alternating voltage is applied across the setup, which further accelerates the particle.
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Lorentz force is the force experienced by electrically charged particles moving in a magnetic field.
- The magnitude of the magnetic force (F) on a charge (q) moving at a speed (v) in a magnetic field of strength B is given by
⇒ F = qvB
EXPLANATION:
- When a charged particle moves perpendicular to the magnetic field (θ = 90∘), it follows a curved path and undergoes circular motion.
- Here, the magnetic force supplies the centripetal force which keeps the particle in a circular motion.
- The centripetal force,
\(⇒ F_C = \frac{mv^2}{r}\)
Where m is the mass of the particle, v is the velocity of the particle, and r is the radius of the circular path traced by the particle.
\(\therefore F = F_C⇒qvB = \frac{mv^2}{r}\)
\(⇒\frac{v}{r} = \frac{qB}{m}\) ----(1)
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Angular velocity of the body is given as,
\(⇒ ω = \frac{v}{r} \)
Where r = radius and ω = 2πf, where f is the frequency.
Substituting (1) in the equation for ω,
\(\Rightarrow \omega = \frac{qB}{m} = 2\pi f\)
\(\Rightarrow f =\frac{qB}{2\pi m}\)
\(\Rightarrow f \propto \frac{1}{m}\)
- The angular frequency of the charged particle in a cyclotron is inversely proportional to its mass.
