Question

An electron in a magnetic field moves counterclockwise on a circle in the \(x y\) -plane, with a cyclotron frequency of \(\omega=1.2 \cdot 10^{12} \mathrm{~Hz}\). What is the magnetic field, \(\vec{B}\) ?

Short answer

Answer: The magnetic field acting on the electron is \(\vec{B} = 6.84 \times 10^{-4} \mathrm{T} \hat{z}\).

Step-by-step solution

Formula for cyclotron frequency

The formula for cyclotron frequency is given by: \(\omega = \frac{qB}{m}\) where \(\omega\) is the cyclotron frequency, \(q\) is the charge of the particle, \(B\) is the magnitude of the magnetic field, and \(m\) is the mass of the particle. For an electron, the charge \(q = -e\) and the mass \(m = m_e\), where \(e = 1.6 \times 10^{-19} \mathrm{C}\) and \(m_e = 9.11 \times 10^{-31} \mathrm{kg}\).

Rearranging the formula to find the magnitude of the magnetic field

Rearrange the formula to solve for the magnitude of the magnetic field \(B\): \(B = \frac{m\omega}{q}\)

Substitute the known values and calculate the magnetic field magnitude

Substitute the given value of \(\omega\) and known values of \(e\) and \(m_e\) into the formula and calculate \(B\): \(B = \frac{(9.11 \times 10^{-31} \mathrm{kg})(1.2 \times 10^{12} \mathrm{s}^{-1})}{1.6 \times 10^{-19} \mathrm{C}}\) After calculating, we get: \(B = 6.84 \times 10^{-4} \mathrm{T}\)

Determine the direction of the magnetic field

Here, as the electron moves in counterclockwise direction, the direction of the magnetic field \(\vec{B}\) must be perpendicular to the plane of its motion (xy-plane), along the positive z-direction. This follows the right-hand rule. So, the magnetic field \(\vec{B}\) is: \(\vec{B} = 6.84 \times 10^{-4} \mathrm{T} \hat{z}\)

Key concepts

Magnetic Field Calculation

When calculating the magnetic field for a moving electron, we start by using the concept of cyclotron frequency. The formula for cyclotron frequency is:

• \( \omega = \frac{qB}{m} \)

This formula helps us understand the relationship between the charge of the particle, the mass of the particle, and the magnetic field. Here, \( \omega \) represents the cyclotron frequency, \( q \) is the charge, \( B \) is the magnitude of the magnetic field, and \( m \) is the mass of the particle.
For an electron, \( q = -e \) and \( m = m_e \), where \( e \) is the elementary charge \( (1.6 \times 10^{-19} \mathrm{C}) \) and the electron mass is \( m_e = 9.11 \times 10^{-31} \mathrm{kg} \).
To find the magnitude of the magnetic field, we rearrange the formula:

• \( B = \frac{m\omega}{q} \)

The known value of \( \omega \) and the constants for \( e \) and \( m_e \) are substituted into this equation to calculate the magnetic field. This calculation gives us the field magnitude, \( B = 6.84 \times 10^{-4} \mathrm{T} \).

Electron Motion in Magnetic Field

Electrons exhibit fascinating behavior when they move through a magnetic field. They travel in a circular pattern due to the Lorentz force acting upon them, which is perpendicular to both the velocity of the electron and the direction of the magnetic field. This results in a circular motion, known as cyclotron motion.
For the given problem, we know the electron moves counterclockwise in the xy-plane. This circle is a result of the continuous perpendicular force that keeps changing the electron's direction without altering its speed.

• The velocity vector of the electron is always tangent to the circle.
• The magnetic field provides the inward centripetal force necessary for circular motion.

This type of motion is typical for charged particles in magnetic fields and is central to the operation of devices such as cyclotrons and magnetic confinement fusion devices.

Right Hand Rule

The Right Hand Rule is a simple yet powerful tool to determine the direction of magnetic fields. It is especially useful when considering the direction of forces in a magnetic field. To visualize the Right Hand Rule:

• Point your thumb in the direction of the charge's velocity (here, you assume a positive charge for rule application).
• Curl your fingers in the direction of the magnetic field.
• Your thumb then points in the force direction exerted by the field on the charge.

Since our problem involves an electron, which has a negative charge, the force direction will actually flip. In this scenario of counterclockwise electron motion in the xy-plane, the magnetic field \( \vec{B} \) is along the positive z-direction based on the Right Hand Rule.
Therefore, using the Right Hand Rule allows us to confidently determine that the magnetic field points out of the plane, matching the solution’s final step. This fundamental rule is instrumental in solving many problems involving electromagnetism.