Question

Consider an electron to be a uniformly dense sphere of charge, with a total charge of \(-e=-1.602 \cdot 10^{-19} \mathrm{C}\), 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

Answer: The expressions for the electron's classical angular momentum of rotation (L), magnetic dipole moment (μ), and gyromagnetic ratio (γₑ) are: 1. Angular momentum (L): $$L = \frac{2}{5}MR^2\omega$$ 2. Magnetic dipole moment (μ): $$\mu = -\frac{e\omega R^2}{2}$$ 3. Gyromagnetic ratio (γₑ): $$\gamma_e = \frac{-5e}{4M}$$

Step-by-step solution

Find the Moment of Inertia

To find the angular momentum (L) of the rotating electron, we need to first find its moment of inertia (I). The moment of inertia depends on the shape and mass distribution of the rotating object, and for a uniform solid sphere, the moment of inertia is given by: $$I = \frac{2}{5}MR^2$$ Where M is the mass of the sphere and R is its radius.

Write the expression for Angular Momentum

The formula for the angular momentum (L) of a rotating object is given by: $$L = I\omega$$ Where ω is the angular frequency of rotation. We can substitute the moment of inertia expression from Step 1 to obtain the angular momentum formula for the electron: $$L = \frac{2}{5}MR^2\omega$$ b) Magnetic Dipole Moment:

Find the current due to rotation

We need to find the current flowing through the rotating electron, which can be calculated by dividing the charge (Q) by the time period (T) of the rotation. Since the angular frequency ω is related to the time period as ω = 2π/T, we can rewrite the time period as T = 2π/ω: $$I = \frac{Q}{T} = \frac{-e}{2\pi/\omega} = -\frac{e\omega}{2\pi}$$

Write the expression for Magnetic Dipole Moment

The formula for magnetic dipole moment (μ) of a circular loop carrying current (I) with area (A) is given by: $$\mu = IA$$ For the electron, we have the area of the sphere (in simplified terms) A = πR², since we are approximating it as a circular loop. Substituting the expressions for current and area, we get the magnetic dipole moment: $$\mu = -\frac{e\omega}{2\pi}(\pi R^2)$$ Simplifying, we get: $$\mu = -\frac{e\omega R^2}{2}$$ c) Gyromagnetic Ratio:

Write the expression for Gyromagnetic Ratio

To find the gyromagnetic ratio (γₑ), we need to find the ratio of magnetic dipole moment (μ) to the angular momentum (L). Dividing the expressions we obtained in parts a) and b), we get: $$\gamma_e = \frac{\mu}{L} = \frac{-\frac{e\omega R^2}{2}}{\frac{2}{5}MR^2\omega}$$

Simplify the expression

By simplifying the equation, we get the gyromagnetic ratio (γₑ): $$\gamma_e = \frac{-5e}{4M}$$

Key concepts

Angular Momentum

Angular momentum is a fundamental physical quantity representing the rotational inertia and velocity of a body. It describes the amount of rotation a body possesses due to its motion about an axis. In the case of a spinning electron modeled as a uniformly dense sphere, the angular momentum (\(L\)) is related to its moment of inertia (\(I\fa-rotate-180)) and its angular frequency (\(\omega\fa-rotate-180)), where \(\omega\) illustrates how rapidly the electron is spinning.

Specifically, the angular momentum is given by the product of the moment of inertia and the angular frequency, mathematically expressed as \(L = I\omega\). For an electron, a crucial particle in the realm of quantum mechanics, angular momentum plays a key role. It not only influences the electron's physical motion but also has significant implications for its magnetic properties and how it interacts with external magnetic fields.

Magnetic Dipole Moment

The magnetic dipole moment (\(\mu\)) of a particle or body is a measure of the strength and orientation of its magnetism. For our spinning electron, imagine it as a tiny bar magnet with a 'north' and 'south' pole created by its spin.

The expression for the magnetic dipole moment due to the spinning charge is derived from the current (\(I\)) it effectively produces and the area (\(A\)) through which this current loops. Hence, \(\mu = IA\). By calculating the current as the charge per unit time (\(I = \frac{-e}{2\pi/\omega}\)), and considering the electron's sphere as a circular loop for simplification, the magnetic dipole moment can be derived. The magnetic moment is vital in determining how the electron interacts with magnetic fields and can affect everything from atomic structure to material properties in solids.

Moment of Inertia

The moment of inertia (\(I\)) is akin to mass in rotational motion. It quantifies the resistance of an object to change in its rotational motion. The distribution of mass in relation to the axis of rotation determines the value of the moment of inertia.

For a solid sphere, the standard formula \(I = \frac{2}{5}MR^2\) is used, where \(M\) is the mass and \(R\) is the radius of the sphere. It is crucial when calculating other rotational properties, such as torque, angular acceleration, and of course, angular momentum. The role of the moment of inertia in our spinning electron scenario is to bridge the gap between the mass distribution and the electron's rotational motion to lead us to the angular momentum.

Electron Charge

Electron charge is a fundamental property of matter, contributing to the electrical force between atoms and molecules. The elementary charge (\(e\)), approximately equal to \(1.602 \times 10^{-19} C\text{-}oulo\text{-}mbs\), is a constant value that represents the charge of a single electron, a crucial constant in physics.

In our exercise, the negative of this constant \(-e\) is used to indicate the charge of the electron. Charge is the source of electromagnetism and is responsible for the electron's ability to generate a magnetic dipole moment when it spins. Understanding the electron charge is fundamental to investigating and understanding the interactions within atoms, the structure of matter, and the principles of electromagnetism as a whole.