Question

Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. What magnetic- field magnitude is required for this transition to be induced by photons with frequency 22.7 MHz?

Short answer

The required magnetic field magnitude is approximately 0.53 T.

Step-by-step solution

Understanding the Relationship

The transition between the spin states of protons in a magnetic field involves energy changes related to the frequency of the photon. The formula that links the energy difference with frequency is given by Planck's equation: \[ \Delta E = h \cdot u \]where \( \Delta E \) is the energy difference, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J} \cdot \text{s}) \), and \( u \) is the frequency.

Zeeman Effect Clarification

The energy difference \( \Delta E \) between the parallel and antiparallel states in a magnetic field \( B \) is given by:\[ \Delta E = 2 \cdot \mu_p \cdot B \]where \( \mu_p \) is the magnetic moment of the proton \( (1.41 \times 10^{-26} \text{ J/T}) \).

Equating the Energy Changes

By equating the expressions from Step 1 and Step 2, we get:\[ h \cdot u = 2 \cdot \mu_p \cdot B \]We can rearrange this formula to solve for the magnetic field \( B \): \[ B = \frac{h \cdot u}{2 \cdot \mu_p} \]

Substituting Known Values

Substitute the known values into the equation:Given \( u = 22.7 \times 10^6 \text{ Hz} \) and Planck's constant \( h = 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \), and \( \mu_p = 1.41 \times 10^{-26} \text{ J/T} \), we find:\[ B = \frac{6.626 \times 10^{-34} \times 22.7 \times 10^6}{2 \times 1.41 \times 10^{-26}} \]

Performing the Calculation

Calculate the magnetic field:\[ B = \frac{6.626 \times 22.7 \times 10^{-28}}{2.82 \times 10^{-26}} \]Solving this gives:\[ B \approx 0.53 \text{ T} \]

Key concepts

Magnetic Field

A magnetic field is a fundamental concept in physics representing a region where a magnetic force acts on charged particles. In this problem, hydrogen atoms placed in an external magnetic field experience effects on their protons. Protons, being positively charged particles, respond to the magnetic field by altering their spin states. Generally, a magnetic field can influence materials and particles in various ways:

• It aligns magnetic dipoles.
• Can induce electric currents in conductors.
• Is quantified by its strength and direction.

Here, the magnetic field induces transitions in the proton's spin states, which we calculate by understanding both energy differences and photon interactions. The stronger the field, the greater the energy differences in spin states.

Proton Spin States

Protons possess an intrinsic property called 'spin,' which is similar to the idea of angular momentum. In the context of this exercise, protons in hydrogen atoms can have their spin aligned either parallel or antiparallel to the external magnetic field:

• Parallel spin state: When the spin of the proton aligns with the magnetic field direction.
• Antiparallel spin state: When the proton's spin is opposite to the direction of the magnetic field.

These different spin orientations correspond to slightly different energy levels. The Zeeman Effect is a crucial phenomenon observing how these energy levels split under the influence of a magnetic field. Transitions between these states involve changes in energy that can be facilitated by absorbing or emitting photons, highlighting how the protons' magnetic moments interact with the field.

Planck's Equation

Planck's equation is foundational in linking energy and frequency in photon-related processes. It is given by the formula:\[ \Delta E = h \cdot u \]where:

• \( \Delta E \) is the change in energy.
• \( h \) is Planck's constant, \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \).
• \( u \) is the frequency of the photon.

In this exercise, the frequency of 22.7 MHz is linked to the required energy change for transitions between proton spin states in the magnetic field. By rearranging Planck’s equation and using known constants such as the magnetic moment, we can derive the necessary magnetic field strength needed for this specific frequency. This demonstrates the interplay of quantum mechanical principles with physical phenomena.