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. Show that its angular momentum is L=qBr² , around the center of the circle.

검증된 단계별 안내

To solve part (a), start by recalling the force acting on a charged particle moving in a magnetic field. The magnetic force provides the centripetal force for circular motion: \( F = qvB = \frac{mv^2}{r} \), where \( v \) is the particle's velocity, \( r \) is the radius of the circular path, \( m \) is the mass, and \( q \) is the charge.

Rearrange the equation \( qvB = \frac{mv^2}{r} \) to solve for the velocity \( v \): \( v = \frac{qBr}{m} \).

The kinetic energy of the particle is given by \( KE = \frac{1}{2}mv^2 \). Substitute \( v = \frac{qBr}{m} \) into this expression: \( KE = \frac{1}{2}m\left(\frac{qBr}{m}\right)^2 \). Simplify to show that \( KE = \frac{q^2B^2r^2}{2m} \), which demonstrates that the kinetic energy is proportional to \( r^2 \).

For part (b), recall the definition of angular momentum \( L \) for a particle moving in a circular path: \( L = mvr \). Substitute \( v = \frac{qBr}{m} \) into this expression: \( L = m\left(\frac{qBr}{m}\right)r \).

Simplify the expression for \( L \): \( L = qBr^2 \). This shows that the angular momentum of the particle is \( L = qBr^2 \), as required.

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주요 개념

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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.

Angular Momentum

Angular momentum (L) is a measure of the rotational motion of an object and is defined as L = r × p, where r is the radius vector and p is the linear momentum. For a charged particle in a magnetic field, the angular momentum can be expressed as L = qBr², where q is the charge and B is the magnetic field strength. This relationship indicates that angular momentum increases with the square of the radius, emphasizing the influence of both charge and magnetic field on rotational dynamics.

Magnetic Force and Circular Motion

When a charged particle moves in a magnetic field, it experiences a magnetic force that acts perpendicular to its velocity, causing it to follow a circular path. The magnetic force can be described by the Lorentz force equation, F = q(v × B). This force provides the necessary centripetal force for circular motion, which is essential for understanding how the radius of curvature and the magnetic field strength affect the particle's motion and energy.