Question

An electron has a spin magnetic moment of magnitude \(\mu=9.285 \cdot 10^{-24} \mathrm{~A} \mathrm{~m}^{2}\). Consequently, it has energy associated with its orientation in a magnetic field. If the difference between the energy of an electron that is "spin up" in a magnetic field of magnitude \(B\) and the energy of one that is "spin down" in the same magnetic field (where "up" and "down" refer to the direction of the magnetic field) is \(9.460 \cdot 10^{-25} \mathrm{~J}\), what is the field magnitude, \(B\) ?

Short answer

Answer: The magnitude of the magnetic field is approximately \(1.019 T\).

Step-by-step solution

Write the Energy Difference Formula

To find the magnetic field magnitude \(B\), we need to use the formula \(\Delta E = \mu \cdot B\), where \(\Delta E\) is the energy difference between spin up and spin down electrons, \(\mu\) is the spin magnetic moment, and \(B\) is the magnetic field magnitude.

Plug in the given values

We are given the energy difference \(\Delta E = 9.460 \cdot 10^{-25}\mathrm{J}\) and the spin magnetic moment \(\mu = 9.285 \cdot 10^{-24}\mathrm{A}\mathrm{m}^{2}\). Substitute these values into the equation to get: \(9.460 \cdot 10^{-25}\mathrm{J} = (9.285 \cdot 10^{-24}\mathrm{A}\mathrm{m}^{2}) \cdot B\)

Solve for the magnetic field magnitude \(B\)

To find \(B\), divide both sides of the equation by the spin magnetic moment \(\mu = 9.285 \cdot 10^{-24}\mathrm{A}\mathrm{m}^{2}\): \(B = \frac{9.460 \cdot 10^{-25}\mathrm{J}}{9.285 \cdot 10^{-24}\mathrm{A}\mathrm{m}^{2}}\)

Calculate the value of the magnetic field magnitude \(B\)

Now, perform the division to find the value of \(B\): \(B = \frac{9.460 \cdot 10^{-25}\mathrm{J}}{9.285 \cdot 10^{-24}\mathrm{A}\mathrm{m}^{2}} = 1.019 \cdot 10^{1} \mathrm{T}\) The magnetic field magnitude, \(B\), is approximately \(1.019 T\).

Key concepts

Magnetic Field Magnitude

In physics, magnetic field magnitude is a measure of the strength of a magnetic field at a given point in space. It's denoted by the symbol B and measured in teslas (T) in the International System of Units (SI). A magnetic field exerts a force on moving electric charges and magnetic dipoles, and it's a vector field, which means it has both a magnitude and a direction.

Understanding magnetic field magnitude is crucial when we examine how charged particles, such as electrons, interact with the field. These interactions are fundamental to numerous technologies, including electric motors, MRI machines, and data storage devices. Moreover, the magnitude of a magnetic field can also affect chemical and physical processes, such as the alignment of atomic spins in the material.

Energy Difference in Magnetic Fields

The energy difference in magnetic fields refers to the change in potential energy of a magnetic moment when its orientation changes relative to the magnetic field. This concept is critical when analyzing phenomena like electron spin in magnetic resonance imaging (MRI) or electron configurations in atoms.

The Zeeman effect, for instance, is a phenomenon in quantum mechanics that occurs when an external magnetic field splits a degenerate energy level due to the interaction with the atomic magnetic moments. The energy difference \(\Delta E\) between the orientations - commonly referred to as 'spin up' and 'spin down' states - depends on the magnetic field magnitude and the intrinsic magnetic moment of the particle, calculated using the formula \(\Delta E = \mu \cdot B\), where \(\mu\) is the magnetic moment.

Spin Magnetic Moment Calculation

The spin magnetic moment is a property of particles that arises from their intrinsic spin and charge. For electrons, the spin magnetic moment is associated with their spin and fundamental charge. It plays a crucial role in determining the behavior of electrons in magnetic fields and is represented by the Greek letter \(\mu\).

To calculate the spin magnetic moment of an electron, physicists use a simplified formula: \(\mu = -g\cdot\frac{e}{2m} \cdot S\), where \(g\) is the g-factor (a dimensionless quantity), \(e\) is the electron charge, \(m\) is the electron mass, and \(S\) is the spin quantum number. However, in most practical cases, the value for an electron's magnetic moment is a constant known as the Bohr magneton, and the calculation involves this predefined constant.

Spin Quantum Number

The spin quantum number, denoted as \(S\), is an intrinsic property of particles like electrons, indicating their angular momentum due to their spin. The concept of 'spin' is a quantum mechanical property, not akin to spinning in the classical sense, but rather a form of angular momentum that doesn't have an equivalent in classical physics.

For electrons, the spin quantum number can take on the values of \(+1/2\) or \(–1/2\), corresponding to 'spin up' and 'spin down' states. These discrete states are a fundamental part of quantum mechanics and are important in understanding everything from the structure of atoms to the principles of quantum computing. The spin of particles has significant implications in magnetic fields, determining the energy levels and the magnetic properties of materials.