Question

Which of the following has the largest cyclotron frequency? a) an electron with speed $v$ in a magnetic field with magnitude $B$ b) an electron with speed $2v$ in a magnetic field with magnitude $B$ c) an electron with speed $v/2$ in a magnetic field with magnitude $B$ d) an electron with speed $2v$ in a magnetic field with magnitude $B/2$ e) an electron with speed $v/2$ in a magnetic field with magnitude $2B$

Short answer

Answer: e) an electron with speed v/2 in a magnetic field with magnitude 2B

Step-by-step solution

Calculate cyclotron frequency for options a, b, c, d, and e

Since the speed of the electron doesn't affect the cyclotron frequency, we only need to focus on the magnetic field magnitude. The given cases can be written as: a) ${\omega }_{c1}=\frac{eB}{m_e}$ b) ${\omega }_{c2}=\frac{eB}{m_e}$ c) ${\omega }_{c3}=\frac{eB}{m_e}$ d) ${\omega }_{c4}=\frac{e(\frac{B}{2})}{m_e}$ e) ${\omega }_{c5}=\frac{e(2B)}{m_e}$

Compare the cyclotron frequencies

Now that we have the cyclotron frequencies for each option, we simply need to compare them to find which has the largest frequency: a) ${\omega }_{c1}=\frac{eB}{m_e}$ b) ${\omega }_{c2}=\frac{eB}{m_e}$ c) ${\omega }_{c3}=\frac{eB}{m_e}$ d) ${\omega }_{c4}=\frac{e(\frac{B}{2})}{m_e}=\frac{eB}{2m_e}$ e) ${\omega }_{c5}=\frac{e(2B)}{m_e}=\frac{2eB}{m_e}$

Determine the case with the largest cyclotron frequency

Comparing the cyclotron frequencies, we can see that ${\omega }_{c5}$ is double the other frequencies and therefore the largest: a) ${\omega }_{c1}=\frac{eB}{m_e}$ b) ${\omega }_{c2}=\frac{eB}{m_e}$ c) ${\omega }_{c3}=\frac{eB}{m_e}$ d) ${\omega }_{c4}=\frac{eB}{2m_e}$ e) ${\omega }_{c5}=\frac{2eB}{m_e}$ (largest cyclotron frequency) Thus, the answer is option e) an electron with speed $v/2$ in a magnetic field with magnitude $2B$.

Key concepts

Magnetic Field

A magnetic field is an invisible force field surrounding magnetic materials and electric currents. It plays a crucial role in the behavior of charged particles like electrons. The magnetic field is represented by the symbol $B$ and is measured in teslas (T).
Magnetic fields can exert forces on moving charged particles, causing them to move in a circular or spiral path. This motion results from the magnetic force acting perpendicular to the direction of motion of the charged particle, influencing its trajectory.

• Magnetic fields are generated by electric currents and magnetic materials
• A charged particle within a magnetic field experiences a force known as the Lorentz force
• The magnitude of this force depends on the charge of the particle, its velocity, and the strength of the magnetic field

Understanding how magnetic fields interact with charged particles like electrons is essential in various technological applications, including cyclotrons and mass spectrometers.

Electron Dynamics

Electron dynamics involves studying the movement and behavior of electrons, especially when they are subjected to forces like magnetic fields. When a magnetic field is applied, electrons move in a circular orbit due to the Lorentz force, which is perpendicular to their velocity. This circular motion is what cyclotrons utilize to accelerate charged particles to high speeds.
In our specific problem, the cyclotron frequency is given by the formula ${\omega }_c=\frac{eB}{m_e}$, where:

• $e$ is the charge of the electron (approximately $1.6\times {10}^{-19}$ Coulombs)
• $B$ is the magnetic field strength
• $m_e$ is the electron mass $(9.11\times {10}^{-31}$ kilograms)

The key concept to grasp here is that the speed of the electron, $v$, does not affect the cyclotron frequency. Only the strength of the magnetic field and the intrinsic properties of the electron matter. This is why options a, b, and c in the given exercise have the same frequency.

Physics Problem Solving

Solving physics problems requires a structured approach to truly understand the underlying principles and apply them effectively. In this particular exercise about cyclotron frequency and magnetic fields, the process can be broken down into a few core steps:
Firstly, it's crucial to identify what factors influence the primary quantity of interest—in this case, the cyclotron frequency. Recognizing that the cyclotron frequency is determined by the charge, mass of the electron, and the magnetic field strength is key.
Secondly, translating the problem to mathematical expressions helps in visualizing and comparing different scenarios. Using the cyclotron frequency formula, calculate the frequency for each scenario to observe differences.
Finally, comparison and analysis of results are crucial. By comparing the resulting expressions for cyclotron frequency from each option, clear patterns emerge, such as recognizing which scenario maximizes this frequency. This kind of logical reasoning and comparison not only solves the problem but also enhances understanding.

• Identify influential factors
• Translate the problem into mathematical language
• Compare results to determine solution

Employing this structured problem-solving method aids in dissecting complex physics problems, making them manageable and comprehensible.