The Magnetic Moment of an Electron

The magnetic moment of a particle is a measure of its ability to respond to a magnetic field. In the case of an electron, its magnetic moment arises from two sources - its intrinsic spin and its orbital motion.

Magnetic Moment (μ) and Angular Momentum (L)

The magnetic moment (μ) of an electron can be related to its angular momentum (L) through the following equation:

μ = γL

where γ is the gyromagnetic ratio, a constant that depends on the properties of the particle.

Orbital Motion and Angular Momentum

When an electron revolves in a circle with uniform speed, it undergoes orbital motion. The angular momentum (L) of the electron is given by:

L = mvr

where m is the mass of the electron, v is its velocity, and r is the radius of the circular path.

Magnetic Moment and Orbital Motion

The magnetic moment (μ) due to the orbital motion of the electron can be calculated using the equation:

μ = γL = γmvr

The gyromagnetic ratio (γ) for an electron is a fundamental constant equal to -e/2m, where e is the charge of the electron and m is its mass.

Magnetic Moment and Angular Momentum Ratio

The ratio of the magnetic moment (μ) to the angular momentum (L) of the electron can be calculated as:

μ/L = γmvr/mvr = γ

Substituting the value of γ for an electron (-e/2m), we get:

μ/L = -e/2m

SI Unit and Decimal Places

The SI unit for the magnetic moment is the ampere-meter squared (A·m^2), and the SI unit for the angular momentum is kilogram meter squared per second (kg·m^2/s). Therefore, the ratio of the magnetic moment to the angular momentum of an electron will have the unit A·m^2 / (kg·m^2/s), which simplifies to A·s/m.

To determine the decimal value, we need to know the specific values of the charge and mass of the electron. The charge of an electron is approximately -1.602 x 10^-19 coulombs, and the mass of an electron is approximately 9.109 x 10^-31 kilograms.

By substituting these values into the ratio equation:

μ/L = -e/2m = (-1.602 x 10^-19 C) / (2 * 9.109 x 10^-31 kg)

Calculating this expression will give us the decimal value of the ratio, which can be rounded off to the appropriate number of decimal places as required.