Deuterons in a cyclotron travel in a circle with radius $32.0$ cm just before emerging from the dees. The frequency of the applied alternating voltage is $9.00$ MHz. Find the magnetic field.

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Step 1: Understand the relationship between the cyclotron frequency and the magnetic field. The cyclotron frequency is given by $ f = \frac{qB}{2\pi m} $, where $ f $ is the frequency, $ q $ is the charge of the particle, $ B $ is the magnetic field, and $ m $ is the mass of the particle. For a deuteron, $ q = 1.6 \times 10^{-19} \, \text{C} $ and $ m = 3.34 \times 10^{-27} \, \text{kg} $.

Step 2: Rearrange the formula to solve for the magnetic field $ B $. The equation becomes $ B = \frac{2\pi m f}{q} $. Substitute the given frequency $ f = 9.00 \, \text{MHz} = 9.00 \times 10^{6} \, \text{Hz} $ into the equation.

Step 3: Perform unit conversions if necessary. Ensure all values are in SI units before substituting them into the formula. For example, the radius $ r $ is given as 32.0 cm, which is $ 0.32 \, \text{m} $, but it is not directly needed for calculating $ B $ in this step.

Step 4: Substitute the values for $ m $, $ f $, and $ q $ into the formula $ B = \frac{2\pi m f}{q} $. This will give the magnetic field $ B $ in teslas (T).

Step 5: Verify the result conceptually. The magnetic field should be consistent with the frequency and the mass-to-charge ratio of the deuteron. If the calculated $ B $ is reasonable, it confirms the setup of the cyclotron is correct.

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주요 개념

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Cyclotron Motion

Cyclotron motion refers to the circular motion of charged particles, such as deuterons, in a magnetic field. The radius of the circular path is determined by the particle's mass, charge, and the strength of the magnetic field. In a cyclotron, particles are accelerated by an alternating voltage, causing them to gain energy and increase their speed while maintaining a circular trajectory.

Magnetic Field and Frequency Relationship

The relationship between the magnetic field strength (B) and the frequency (f) of a charged particle in a cyclotron is given by the equation f = (qB)/(2πm), where q is the charge, m is the mass, and B is the magnetic field. This equation shows that the frequency of the alternating voltage must match the cyclotron frequency for effective acceleration of the particles. Understanding this relationship is crucial for calculating the magnetic field strength.

Radius of Circular Motion

The radius of the circular path of a charged particle in a magnetic field is influenced by its momentum and the magnetic field strength. The formula r = (mv)/(qB) relates the radius (r) to the mass (m), velocity (v), charge (q), and magnetic field (B). This concept is essential for determining the magnetic field strength when the radius and other parameters are known, as in the case of the deuterons in the cyclotron.