Question

Show that a charged particle in a uniform magnetic field describes a circular orbit with an orbital period that is independent of the radius. Find the relationship between the angular velocity and magnetic flux density for an electron (the cyclotron frequency).

Short answer

Short Answer: The orbital period of a charged particle in a uniform magnetic field is independent of the radius, as shown by the expression T = 2 * π * m / (q * B), where T is the orbital period, m is the mass of the particle, q is its charge, and B is the magnetic flux density. The cyclotron frequency for an electron in a uniform magnetic field is given by ω = (e * B) / m_e, where ω is the angular velocity, e is the elementary charge, and m_e is the mass of the electron.

Step-by-step solution

Determine the force on the charged particle in a magnetic field

To determine the force acting on a charged particle in a magnetic field, we use the Lorentz force equation: F = q(v × B) where F is the force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field vector. Since the magnetic field is uniform in this case, the force will always be perpendicular to the velocity vector.

Calculate centripetal acceleration

The force acting on the charged particle causes it to move in a circular orbit. The centripetal acceleration is given by the following equation: a_c = (v^2) / r where a_c is the centripetal acceleration, v is the speed of the particle, and r is the radius of its orbit.

Equate the Lorentz force to the centripetal force

The Lorentz force acting on the charged particle is responsible for keeping it in the circular orbit. Therefore, we can equate the magnitude of the Lorentz force to the centripetal force (F_c = m * a_c): |q(v × B)| = m * (v^2) / r Since the force is perpendicular to the velocity vector, we can further simplify this equation as: q * v * B = m * (v^2) / r

Derive the orbital period

The orbital period T is the time it takes for the charged particle to complete one full orbit. Since the path is circular, we can relate the orbital period to the speed and radius as follows: v = (2 * π * r) / T

Calculate the independence of the orbital period on the radius

Next, we can substitute the expression for v from Step 4 into the equation from Step 3 to eliminate the speed term: q * ((2 * π * r) / T) * B = m * (((2 * π * r) / T)^2) / r Solving for T, we have: T = 2 * π * m / (q * B) This expression shows that the orbital period T is independent of the radius r, as required.

Derive the cyclotron frequency

The angular velocity ω is related to the orbital period by: ω = 2 * π / T Using the expression for T from Step 5, we can derive the relationship between the angular velocity and magnetic flux density: ω = 2 * π / (2 * π * m / (q * B)) ω = (q * B) / m For an electron, q = -e, and therefore: ω = (e * B) / m_e where e is the elementary charge, and m_e is the mass of the electron. This expression gives the cyclotron frequency for an electron in a uniform magnetic field.

Key concepts

Lorentz Force

The Lorentz force plays a critical role in the behavior of charged particles in magnetic fields. Essentially, it is the combination of electric and magnetic forces on a point charge due to electromagnetic fields. However, in the scenario of a charged particle such as an electron moving perpendicular to a uniform magnetic field, only the magnetic component is considered, and the force can be described by the equation:
\( F = q(\textbf{v} \times \textbf{B}) \).
Here, \( F \) represents the Lorentz force, \( q \) is the charge of the particle, \( \textbf{v} \) is the particle's velocity vector, and \( \textbf{B} \) is the magnetic field vector. The cross product indicates that the force is always perpendicular to both the velocity and the magnetic field, resulting in the charged particle moving in a circular path. This force is responsible for providing the necessary centripetal force to keep the particle in circular motion.

Centripetal Acceleration

When a charged particle moves in a circle under the influence of the Lorentz force, it experiences an inward acceleration called centripetal acceleration, which can be calculated using the formula:
\( a_c = \frac{v^2}{r} \),
where \( a_c \) is the centripetal acceleration, \( v \) is the particle's velocity, and \( r \) is the radius of the circular path. It's essential to understand that this acceleration does not change the particle's speed but rather its direction. Centripetal acceleration is always directed towards the center of the circle and maintains the particle's circular motion. By applying Newton's second law, \( F_c = m \cdot a_c \), we can relate the Lorentz force to the centripetal force needed to keep the particle on its circular path.

Magnetic Field

The magnetic field, denoted by \( \textbf{B} \), is a vector field that exerts a force on moving charges, and it is essential to the concept of cyclotron motion. Its strength and direction determine how the charged particles will move when they enter the field. Uniform magnetic fields, like the one considered in our exercise, have the same strength and direction at every point within the region. This consistency is crucial for the circular path that the charged particles exhibit when subjected to these fields due to the steady Lorentz force. The interaction between the charged particle and the magnetic field can be used to manipulate the particle's trajectory for various applications, such as in a cyclotron, a device that accelerates charged particles to high energies.

Angular Velocity

Angular velocity, \( \omega \), is a scalar measure of the rotation rate of an object as it revolves around a central point. In the context of a charged particle moving in a magnetic field, angular velocity specifically refers to how fast the particle revolves around the center of its circular path. It is related to the particle's orbital period \( T \), which is the time taken for one complete revolution, by the equation:
\( \omega = \frac{2\pi}{T} \).
By understanding that the orbital period is independent of the particle's radius of motion as shown in our exercise, we can conclude that the angular velocity for a given charge and magnetic field is also a constant. This constant is known as the cyclotron frequency for charged particles moving in a uniform magnetic field, and it underscores the principle behind the operation of cyclotrons, particle accelerators that utilize magnetic fields to control particle motion.