Find the energy needed to flip the spin state of a proton in Earth's magnetic field, whose magnitude is about \(30 \mu \mathrm{T}.\)

Understanding the proton's magnetic moment is essential because it is a key factor in calculating the energy needed to alter a proton's spin state. Fundamentally, magnetic moment is a measure of the amount of magnetic force an object can exert on an electric current or how much it will twist in a magnetic field. Protons, just like tiny bar magnets, have a positive charge and spin, which together contribute to their magnetic moment.

The magnetic moment of a proton is quantified using a mathematical equation:

\( \mu = g \cdot \frac{e \cdot \hbar}{2m} \)

where:

• \(g\) is the g-factor for a proton, approximately 5.586, demonstrating how the proton's magnetic moment compares to its theoretical value.
• \(e\) is the elementary charge, which is the charge of a proton, measured as approximately \(1.602 \times 10^{-19}\) Coulombs.
• \(\hbar\) is the reduced Planck's constant – a fundamental number in quantum mechanics representing the relation between a photon's energy and its frequency, at about \(1.055 \times 10^{-34}\) m\(^2\) kg/s.
• \(m\) is the mass of the proton, roughly \(1.673 \times 10^{-27}\) kg.

Calculating this gives us a numerical value of the proton's magnetic moment, which is directly used in calculating the energy needed to flip its spin in a magnetic field, such as Earth's.

The Earth is surrounded by a magnetic field, similar to the way a bar magnet generates a field. This field plays a crucial role in the functionality of compasses and the protection of our planet from solar wind. While this field is not uniform and varies in different locations, over Earth's surface, it is approximately 30 microteslas (µT) in magnitude.

Understanding the Earth's magnetic field intensity is necessary when calculating the effect of this field on particles like protons. In the context of our discussion, we consider the Earth's magnetic field as a uniform field affecting a proton's spin. This simplification allows us to use Earth's magnetic field as a constant value in our calculations, specifically when we use it in conjunction with the magnetic moment of the proton to determine the energy needed for a spin flip.

The conservation of energy is a fundamental principle of physics that states energy cannot be created or destroyed, only transformed from one form to another. This principle is instrumental in understanding how energy is involved when we talk about flipping the spin of a proton.

When a proton flips its spin, energy must be absorbed or released to accommodate this change. To calculate this energy, we use an adapted conservation of energy equation that describes the work done to flip the proton's spin in a magnetic field, which is:

\( E = -\mu \cdot B \cdot \cos(\theta) \)

where:

• \(E\) is the energy required to flip the spin.
• \(\mu\) is the magnetic moment of the proton.
• \(B\) is the magnetic field strength - in our case, Earth's magnetic field.
• \(\theta\) is the angle between the proton's magnetic moment and the magnetic field, which for a complete flip from one state to the opposite is 180 degrees – making \(\cos(\theta) = -1\).

Applying this formula, we understand that to flip the spin, the proton absorbs energy equivalent to the product of its magnetic moment and the magnetic field strength.