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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

An electron is moving at \(v=6.00 \cdot 10^{7} \mathrm{~m} / \mathrm{s}\) perpendicular to the Earth's magnetic field. If the field strength is \(0.500 \cdot 10^{-4} \mathrm{~T}\), what is the radius of the electron's circular path?

An alpha particle \(\left(m=6.6 \cdot 10^{-27} \mathrm{~kg}, q=+2 e\right)\) is accelerated by a potential difference of \(2700 \mathrm{~V}\) and moves in a plane perpendicular to a constant magnetic field of magnitude \(0.340 \mathrm{~T}\), which curves the trajectory of the alpha particle. Determine the radius of curvature and the period of revolution.

A rail gun accelerates a projectile from rest by using the magnetic force on a current-carrying wire. The wire has radius \(r=5.1 \cdot 10^{-4} \mathrm{~m}\) and is made of copper having a density of \(\rho=8960 \mathrm{~kg} / \mathrm{m}^{3}\). The gun consists of rails of length \(L=1.0 \mathrm{~m}\) in a constant magnetic field of magnitude \(B=2.0 \mathrm{~T}\) oriented perpendicular to the plane defined by the rails. The wire forms an electrical connection across the rails at one end of the rails. When triggered, a current of \(1.00 \cdot 10^{4}\) A flows through the wire, which accelerates the wire along the rails. Calculate the final speed of the wire as it leaves the rails. (Neglect friction.)

An electron (with charge \(-e\) and mass \(m_{\mathrm{e}}\) ) moving in the positive \(x\) -direction enters a velocity selector. The velocity selector consists of crossed electric and magnetic fields: \(\vec{E}\) is directed in the positive \(y\) -direction, and \(\vec{B}\) is directed in the positive \(z\) -direction. For a velocity \(v\) (in the positive \(x\) -direction \(),\) the net force on the electron is zero, and the electron moves straight through the velocity selector. With what velocity will a proton (with charge \(+e\) and mass \(m_{\mathrm{p}}=1836 \mathrm{~m}_{\mathrm{e}}\) ) move straight through the velocity selector? a) \(v\) b) \(-v\) c) \(v / 1836\) d) \(-v / 1836\)

A straight wire with a constant current running through it is in Earth's magnetic field, at a location where the magnitude is \(0.43 \mathrm{G}\). What is the minimum current that must flow through the wire for a 10.0 -cm length of it to experience a force of \(1.0 \mathrm{~N} ?\)
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