(II) For a particle of mass m and charge q moving in a circular path in a magnetic field B, (a) show that its kinetic energy is proportional to r² , the square of the radius of curvature of its path. (b) Show that its angular momentum is L=qBr² , around the center of the circle.

Verified step by step guidance

1. Identify the force acting on the charged particle: The magnetic force acting on a charged particle moving in a magnetic field is given by the Lorentz force, which is F = qvB, where q is the charge, v is the velocity, and B is the magnetic field strength. This force provides the centripetal force required for circular motion.
2. Relate the magnetic force to the centripetal force: For circular motion, the centripetal force F_c is given by F_c = \frac{mv^2}{r}, where m is the mass, v is the velocity, and r is the radius of the circle. Since the magnetic force acts as the centripetal force, set qvB = \frac{mv^2}{r}.
3. Express kinetic energy in terms of r: Solve the equation qvB = \frac{mv^2}{r} for v^2 and substitute it into the expression for kinetic energy, KE = \frac{1}{2}mv^2. This will show how KE is related to r².
4. Derive the expression for angular momentum: Angular momentum L for a particle in circular motion is given by L = mvr. Use the relationship between v and r derived from the magnetic force equation to express L in terms of q, B, and r².
5. Verify the proportionality and expressions: Check the derived expressions to ensure they show the correct proportionality (KE ∝ r²) and the correct formula for angular momentum (L = qBr²).

Key Concepts

Kinetic Energy in Circular Motion

In circular motion, the kinetic energy (KE) of a particle is given by the formula KE = (1/2)mv², where m is the mass and v is the velocity. For a charged particle moving in a magnetic field, the velocity can be related to the radius of curvature (r) and the magnetic field strength (B). As the radius increases, the velocity increases, leading to a proportional increase in kinetic energy, specifically showing that KE is proportional to r².

Magnetic Force and Circular Motion

When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as a centripetal force, keeping it in circular motion. This force is given by F = qvB, where q is the charge, v is the velocity, and B is the magnetic field strength. The balance between this magnetic force and the required centripetal force (F = mv²/r) allows us to derive relationships involving the radius of curvature and the particle's velocity.

Angular Momentum in Magnetic Fields

Angular momentum (L) for a particle moving in a circular path is defined as L = r × p, where p is the linear momentum (p = mv). In the context of a charged particle in a magnetic field, the angular momentum can also be expressed as L = qBr², indicating that it is directly proportional to the charge, the magnetic field strength, and the square of the radius of curvature. This relationship highlights how magnetic fields influence the rotational dynamics of charged particles.