The radius of a charged particle moving in a magnetic field is determined by the balance between the magnetic force acting on the particle and the centrifugal force that keeps the particle in circular motion. The magnetic force acting on a charged particle is proportional to the strength of the magnetic field, the charge of the particle, and the velocity of the particle. The centrifugal force, on the other hand, is proportional to the mass of the particle, the velocity of the particle, and the radius of the circular path.
When a larger magnetic field is applied to a charged particle, the magnetic force acting on the particle increases, causing the particle to curve more sharply. This means that the radius of the circular path decreases since the centrifugal force required to keep the particle in circular motion must increase to balance the stronger magnetic force. Mathematically, we can see this effect by considering the equation for the magnetic force:
Fm = qvB
where Fm is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the strength of the magnetic field. The force required to keep the particle in circular motion is given by:
Fc = mv^2 / r
where Fc is the centrifugal force, m is the mass of the particle, v is the velocity of the particle, and r is the radius of the circular path. By equating these two forces, we can solve for the radius of the circular path:
mv^2 / r = qvB
r = mv / qB
From this equation, we can see that the radius of the circular path is inversely proportional to the strength of the magnetic field. Therefore, when a larger magnetic field is applied, the radius of the circular path decreases.