What strength of magnetic field is used in a cyclotron in which protons make 3.5 x 10⁷ revolutions per second?

Verified step by step guidance

1. Understand the problem: A cyclotron accelerates charged particles like protons using a magnetic field. The frequency of revolution (f) is given as 3.5 × 10⁷ revolutions per second. We need to find the magnetic field strength (B). The relationship between the cyclotron frequency and the magnetic field is derived from the equation for circular motion and the Lorentz force.
2. Recall the formula for the cyclotron frequency: \( f = \frac{qB}{2\pi m} \), where \( f \) is the frequency, \( q \) is the charge of the proton (\( 1.6 \times 10^{-19} \; C \)), \( B \) is the magnetic field strength, and \( m \) is the mass of the proton (\( 1.67 \times 10^{-27} \; kg \)).
3. Rearrange the formula to solve for \( B \): \( B = \frac{2\pi m f}{q} \). This equation allows us to calculate the magnetic field strength once we substitute the known values.
4. Substitute the known values into the equation: \( m = 1.67 \times 10^{-27} \; kg \), \( f = 3.5 \times 10^7 \; Hz \), and \( q = 1.6 \times 10^{-19} \; C \). The equation becomes \( B = \frac{2\pi (1.67 \times 10^{-27}) (3.5 \times 10^7)}{1.6 \times 10^{-19}} \).
5. Simplify the expression to calculate \( B \). Perform the multiplication and division step by step to find the magnetic field strength. Ensure the units are consistent and the result is in Tesla (T).

Key Concepts

Cyclotron

A cyclotron is a type of particle accelerator that uses a magnetic field to bend the path of charged particles, such as protons, into a spiral. The particles are accelerated by an alternating electric field as they move in circular paths, gaining energy with each revolution. The frequency of revolution is crucial for determining the magnetic field strength required for the cyclotron's operation.

Magnetic Field Strength

Magnetic field strength, often denoted as B, is a measure of the magnetic force experienced by a charged particle moving through the field. In a cyclotron, the strength of the magnetic field is directly related to the radius of the particle's path and its velocity. The relationship is given by the equation B = (mv)/(qR), where m is mass, v is velocity, q is charge, and R is the radius of the circular path.

Frequency of Revolution

The frequency of revolution refers to how many times a charged particle completes a full circular path in a given time, measured in revolutions per second (Hz). In the context of a cyclotron, this frequency is essential for calculating the required magnetic field strength to maintain the particle's circular motion. The frequency is influenced by the particle's mass, charge, and the magnetic field strength.