Physics Friday 66 « Twisted One 151’s Weblog

Physics Friday 66

By twistedone151

Last time, we showed that a plane loop carrying current I placed in a uniform magnetic field experiences a torque $\mathbf{\tau}=\mathbf{\vec{\mu}}\times\mathbf{\vec{B}}=I\mathbf{\vec{A}}\times\mathbf{\vec{B}}$ , where $\mathbf{\vec{\mu}}=I\mathbf{\vec{A}}$ is the magnetic moment of the loop, and $\mathbf{\vec{A}}$ is the area vector for the loop, as given using the right-hand rule.

Now, let us consider an electron moving in a circular orbit of radius r and angular velocity ω₀. Then the period of the orbit is $T=\frac{2\pi}{\omega_0}$ , and so the charge passing through any point on the orbit per unit time is $\frac{-e}{T}=\frac{-e\omega_0}{2\pi}$ , and so we can treat the circular orbit as a current loop of current $I=-\frac{e\omega_0}{2\pi}$ (the minus sign indicates that the current is opposite in direction to the motion of the electron, as the electron has a negative charge). Thus, the orbit has a magnetic moment, $\mathbf{\vec{\mu}}=I\mathbf{\vec{A}}=-\frac{er^2}{2}\mathbf{\vec{\omega}_0}$ .

Unlike the previous case of a current loop, here we also have to consider the angular momentum of our electron’s orbit as well. As our orbit is circular, the angular momentum is just $\mathbf{\vec{L}}=m_er^2\mathbf{\vec{\omega}_0}$ , where m_e is the mass of the electron. Thus, we can rewrite the magnetic moment in terms of the angular momentum:
$\mathbf{\vec{\mu}}=-\frac{e}{2m_e}\mathbf{\vec{L}}$ .
and the magnetic moment is proportional to the angular momentum of the orbit, with the constant of proportionality dependent only on the properties of the electron.

Note here that since the magnetic moment is proportional to the angular momentum, the torque due to an external magnetic field, $\mathbf{\tau}=\mathbf{\vec{\mu}}\times\mathbf{\vec{B}}$ is perpendicular to both $\mathbf{\vec{L}}$ and $\mathbf{\vec{B}}$ . Thus, as seen in previous work, the orbit, if not perpendicular to the magnetic field, will precess about the field. This is an example of Larmor precession, which occurs whenever there is a magnetic moment proportional to angular momentum exposed to an external magnetic field. The frequency of this precession, the Larmor frequency, for this problem is $\omega_{L}=\frac{d\phi}{dt}=\frac{e}{2m_e}B$ . In the more general form, the constant of proportionality between the magnetic moment and angular momentum is called the gyromagnetic ratio (or sometimes the magnetogyric ratio), and usually denoted by γ: $\mathbf{\vec{\mu}}=\gamma\mathbf{\vec{L}}$ . Then the Larmor frequency is $\omega_{L}=\gamma{B}$ . These frequencies, when applied to a charged particle with spin, are important in spin transitions, and play an important role in systems such as nuclear magnetic resonance

Tags: Electricity & Magnetism, Electron, Friday Physics, Larmor Frequency, Larmor Precession, Magnetic Moment, physics, Precession, Torque

This entry was posted on April 3, 2009 at 1:20 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

One Response to “Physics Friday 66”

Physics Friday 77 « Twisted One 151’s Weblog Says:
June 19, 2009 at 12:08 am | Reply
[...] moment involves a g-factor, and thus a gyromagnetic ratio, incompatible with classic physics (see here). Another important difference is in the allowed values of s. As we found here, m ranges in [...]

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Physics Friday 66

One Response to “Physics Friday 66”

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