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Chapter 27: Problem 3

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 \(2 v\) 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 \(2 v\) in a magnetic field with magnitude \(B / 2\) e) an electron with speed \(v / 2\) in a magnetic field with magnitude \(2 B\)

Short Answer

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

Step by step solution

01

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) \(\omega_{c1} = \frac{e B}{m_e}\) b) \(\omega_{c2} = \frac{e B}{m_e}\) c) \(\omega_{c3} = \frac{e B}{m_e}\) d) \(\omega_{c4} = \frac{e (\frac{B}{2})}{m_e}\) e) \(\omega_{c5} = \frac{e (2B)}{m_e}\)
02

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) \(\omega_{c1} = \frac{e B}{m_e}\) b) \(\omega_{c2} = \frac{e B}{m_e}\) c) \(\omega_{c3} = \frac{e B}{m_e}\) d) \(\omega_{c4} = \frac{e (\frac{B}{2})}{m_e} = \frac{e B}{2m_e}\) e) \(\omega_{c5} = \frac{e (2B)}{m_e} = \frac{2e B}{m_e}\)
03

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) \(\omega_{c1} = \frac{e B}{m_e}\) b) \(\omega_{c2} = \frac{e B}{m_e}\) c) \(\omega_{c3} = \frac{e B}{m_e}\) d) \(\omega_{c4} = \frac{e B}{2m_e}\) e) \(\omega_{c5} = \frac{2e B}{m_e}\) (largest cyclotron frequency) Thus, the answer is option e) an electron with speed \(v / 2\) in a magnetic field with magnitude \(2 B\).

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Most popular questions from this chapter

A particle with mass \(m\), charge \(q\), and velocity \(v\) enters a magnetic field of magnitude \(B\) and with direction perpendicular to the initial velocity of the particle. What is the work done by the magnetic field on the particle? How does this affect the particle's motion?

A charged particle is moving in a constant magnetic field. State whether each of the following statements concerning the magnetic force exerted on the particle is true or false? (Assume that the magnetic field is not parallel or antiparallel to the velocity.) a) It does no work on the particle. b) It may increase the speed of the particle. c) It may change the velocity of the particle. d) It can act only on the particle while the particle is in motion. e) It does not change the kinetic energy of the particle.

A high electron mobility transistor (HEMT) controls large currents by applying a small voltage to a thin sheet of electrons. The density and mobility of the electrons in the sheet are critical for the operation of the HEMT. HEMTs consisting of AlGaN/GaN/Si are being studied because they promise better performance at higher powers, temperatures, and frequencies than conventional silicon HEMTs can achieve. In one study, the Hall effect was used to measure the density of electrons in one of these new HEMTs. When a current of \(10.0 \mu\) A flows through the length of the electron sheet, which is \(1.00 \mathrm{~mm}\) long, \(0.300 \mathrm{~mm}\) wide, and \(10.0 \mathrm{nm}\) thick, a magnetic field of \(1.00 \mathrm{~T}\) perpendicular to the sheet produces a voltage of \(0.680 \mathrm{mV}\) across the width of the sheet. What is the density of electrons in the sheet?

A rectangular coil with 20 windings carries a current of 2.00 mA flowing in the counterclockwise direction. It has two sides that are parallel to the \(y\) -axis and have length \(8.00 \mathrm{~cm}\) and two sides that are parallel to the \(x\) -axis and have length \(6.00 \mathrm{~cm} .\) A uniform magnetic field of \(50.0 \mu \mathrm{T}\) acts in the positive \(x\) -direction. What torque must be applied to the loop to hold it steady?

The work done by the magnetic field on a charged particle in motion in a cyclotron is zero. How, then, can a cyclotron be used as a particle accelerator, and what essential feature of the particle's motion makes it possible? A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One \(\bullet\) and two \(\bullet\) indicate increasing level of problem difficulty.
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