Question

Consider an electron to be a uniformly dense sphere of charge, with a total charge of \(-e=-1.60 \cdot 10^{-19} \mathrm{C}\) spinning at an angular frequency, \(\omega\). a) Write an expression for its classical angular momentum of rotation, \(L\) b) Write an expression for its magnetic dipole moment, \(\mu\). c) Find the ratio, \(\gamma_{e}=\mu / L\), known as the gyromagnetic ratio.

Short answer

a) The expression for the classical angular momentum of rotation, \(L\), is: \(L=I\omega=\frac{2}{5}m\left(\frac{3Q}{4\pi\rho}\right)^\frac{2}{3}\omega\) b) The expression for the magnetic dipole moment, \(\mu\), is: \(\mu=\frac{1}{2}Q\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}\omega\) c) The expression for the gyromagnetic ratio, \(\gamma_e\), is: \(\gamma_e=\frac{5}{2}\frac{m}{\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}}\)

Step-by-step solution

Find the classical angular momentum of rotation L

For a rotating solid sphere, the angular momentum can be found using the formula: \(L=I\omega\) where \(I\) is the moment of inertia and \(\omega\) is the angular frequency. For a solid sphere of mass \(m\) and radius \(r\), the moment of inertia is given by: \(I=\frac{2}{5}mr^2\) However, we are not given the mass nor the radius of the electron. Instead, we are given its charge and told that it's uniformly dense. We can relate the mass to the charge by density, noting that for a sphere of charge density \(\rho\) and volume \(V\): \(Q=\rho V\) where \(Q = -e\) is the total charge and \(V = \frac{4}{3}\pi r^3\) is the volume of the sphere. We can rearrange this to find the radius in terms of the charge density: \(r=\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}\) Now we can write the moment of inertia in terms of charge density: \(I=\frac{2}{5}m\left(\frac{3Q}{4\pi\rho}\right)^\frac{2}{3}\) Finally, we can write the angular momentum: \(L=I\omega=\frac{2}{5}m\left(\frac{3Q}{4\pi\rho}\right)^\frac{2}{3}\omega\)

Find the magnetic dipole moment μ

For a spinning charged sphere with total charge Q and angular frequency \(\omega\), the magnetic dipole moment can be found using the formula: \(\mu=\frac{1}{2}QR\omega\) where \(R\) is the effective radius (since it's uniformly distributed, this is just the radius \(r\) of the sphere). Using the expression for \(r\) found in step 1, we get: \(\mu=\frac{1}{2}Q\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}\omega\)

Find the gyromagnetic ratio γₑ=μ/L

To find the gyromagnetic ratio, we need to divide the magnetic dipole moment \(\mu\) by the angular momentum \(L\): \(\gamma_e=\frac{\mu}{L}=\frac{\frac{1}{2}Q\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}\omega}{\frac{2}{5}m\left(\frac{3Q}{4\pi\rho}\right)^\frac{2}{3}\omega}\) The angular frequency \(\omega\) and charge \(Q\) cancel out in the ratio, giving: \(\gamma_e=\frac{5}{2}\frac{m}{\left(\frac{3Q}{4\pi\rho}\right)^\frac{1}{3}}\) This is the expression for the gyromagnetic ratio of the spinning electron.

Key concepts

Angular Momentum

Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. For a classical object, such as a spinning sphere, this is quantified as a product of its rotational inertia and its angular velocity. Think of angular momentum as the rotational counterpart to linear momentum, which keeps the object in constant motion. To calculate angular momentum, we use the formula:

• \( L = I \omega \)

Here, \(L\) represents the angular momentum, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity or frequency with which the object rotates. The moment of inertia \(I\) is essentially a measure of how mass is distributed about the axis of rotation, affecting how easily an object spins. For a solid sphere, this is specifically defined by:

• \( I = \frac{2}{5} m r^2 \)

This formula highlights that the distribution of mass (as a function of the sphere's radius \(r\) and mass \(m\)) directly influences its angular momentum. Understanding angular momentum helps us analyze the behavior of rotating systems, like planets or electrons spinning on their axis.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that expresses the torque experienced by a magnetic entity in an external magnetic field. For spinning charged objects, like electrons, the magnetic dipole moment captures how their rotation generates a magnetic field. Electrically charged spinning spheres exhibit this property, leading to a phenomenon much like a tiny magnet, where we can use the formula:

• \( \mu = \frac{1}{2} Q R \omega \)

Here, \(\mu\) is the magnetic dipole moment, \(Q\) is the total charge, \(R\) is the effective radius, and \(\omega\) is the angular frequency. This formula indicates that the charge and how fast it spins affect the strength of its magnetic moment. By utilizing an electron's charge \(-e\) and the expressions derived for its effective radius, we can uniquely characterize its contribution to magnetism in the atomic world. This is crucial in understanding electromagnetic interactions at the quantum level.

Electron Charge

An electron's charge is a fundamental property of its nature, represented by \(-e\), where \(e = 1.60 \times 10^{-19} \text{C}\). This specific quantity not only defines the electron's interactions with electric and magnetic fields but also plays a critical role in determining its behavior in atoms and molecules. Electrons, being negatively charged, are attracted to positively charged particles, enabling their participation in forming chemical bonds and conducting electricity.

• The negative sign indicates that the charge is opposite to positive charge, commonly found in protons.
• Understanding electron charge is essential for grasping the fundamentals of electromagnetism.

Considered one of the building blocks of matter, the charge of an electron influences a vast range of phenomena, from simple circuits to complex chemical reactions, and plays a significant role in the studies of atomic structure and quantum mechanics. The electron charge, together with the concepts of angular momentum and magnetic dipole moments, is foundational in physics, particularly when exploring the properties of microscopic particles like electrons.