A proton and an particle are projected with the same kinetic...

A proton and an particle are projected with the same kinetic energy at right angles to the uniform mag. field. Which one of the following statements will be true. (a) The $\alpha$ - particle will be bent in a circular path with a small radius that for the proton. (b) The radius of the path of the $\alpha$ - particle will be greater than that of the proton. (c) The $\alpha$ - particle and the proton will be bent in a circular path with the same radius. (d) The $\alpha$ - particle and the proton will go through the field in a straight line.

A proton and an particle are projected with the same kinetic energy at right angles to the uniform mag. field. Which one of the following statements will be true.
(a) The $\alpha$ - particle will be bent in a circular path with a small radius that for the proton.
(b) The radius of the path of the $\alpha$ - particle will be greater than that of the proton.
(c) The $\alpha$ - particle and the proton will be bent in a circular path with the same radius.
(d) The $\alpha$ - particle and the proton will go through the field in
a straight line.

Key Concepts

Lorentz Force in a Magnetic Field

Charged particles moving in a magnetic field experience a force known as the Lorentz force, which acts perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for the circular motion of the particle when projected at right angles to the magnetic field, as it provides the necessary centripetal force for circular motion.

Relationship Between Kinetic Energy and Velocity

The kinetic energy of a particle is given by the equation KE = ½ m v². For particles with the same kinetic energy, their velocities will depend on their masses; specifically, a particle with a larger mass will have a lower velocity than one with a smaller mass when both have the same kinetic energy. This relationship is key to understanding the subsequent effect on the radius of their circular paths in a magnetic field.

Determination of the Radius of Circular Motion

The radius of the circular path for a charged particle moving in a magnetic field can be derived from balancing the magnetic (Lorentz) force and the centripetal force required for circular motion. The radius is given by the formula r = m v / (q B), where m is the mass, v is the velocity, q is the charge, and B is the magnetic field strength. This formula shows that the radius depends on both the mass and the charge of the particle, as well as its velocity.

Charge-to-Mass Ratio

The charge-to-mass ratio (q/m) is a crucial parameter in determining the motion of a charged particle in a magnetic field. Even if two particles have the same kinetic energy, differences in their charge-to-mass ratios will affect the radius of their circular trajectories. The radius of the path is, in fact, directly proportional to the square root of the mass and inversely proportional to the charge, highlighting why this ratio is significant in understanding and comparing the motion of different particles under the influence of a magnetic field.

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