Problem 49 - Physics for Scientists and Engineers with Modern Physics

Question

(II) In a Stern-Gerlach experiment, Ag atoms exit the oven with an average speed of \(780 \mathrm{~m} / \mathrm{s}\) and pass through a magnetic field gradient \(d B / d z=1.8 \times 10^{3} \mathrm{~T} / \mathrm{m}\) for a distance of \(5.0 \mathrm{~cm} .(a)\) What is the separation of the two beams as they emerge from the magnet? \((b)\) What would the separation be if the \(g\) -factor were 1 for electron spin?

Step-by-Step Solution

Answer

The separation with the normal \(g\)-factor is given by the product of deflection. With \(g = 1\), the separation reduces accordingly.

Step 1: Understanding the Problem

We are asked to find the separation of two diverging beams of silver (Ag) atoms due to the Stern-Gerlach experiment. This phenomenon occurs because of the magnetic moment of the atoms interacting with the magnetic field gradient. The atoms split based on their spin orientations, and we will calculate the separation of these paths.

Step 2: Calculating Force on Ag Atoms

The force on an atom due to the magnetic field gradient is given by \[ F = \mu_B \frac{dB}{dz} \]where \(\mu_B\) is the magnetic moment of the silver atom. For an electron, this is the Bohr magneton, \(\mu_B = \frac{e\hbar}{2m_e}\). Insert \( g = 2 \) for electron spin and natural constants to calculate \(\mu_B\). Then use \(\frac{dB}{dz} = 1.8 \times 10^3 \text{ T/m}\) to find the force.

Step 3: Determining Deflection Distance

The deflection distance \( y \) can be found by considering the motion as uniformly accelerated motion due to the force:\[ y = \frac{1}{2} a t^2 \]where \(a = \frac{F}{m_{Ag}}\) is the acceleration and \(t\) is the time spent in the magnetic field. Calculate \(t = \frac{L}{v}\), where \(L = 0.05 \text{ m} \) and \(v = 780 \text{ m/s}\). Find \(a\), then substitute \(t\) and \(a\) into the deflection formula to get \( y \). Finally, the total separation is \( 2y \).

Step 4: Considering g-Factor of 1

If the \(g\)-factor were 1, this would modify the effective magnetic moment to \(\mu' = \frac{e\hbar}{4m_e}\). Repeat the calculation for force and acceleration using this modified magnetic moment. Use the same steps to find the new separation \(2y'\).

Key Concepts

Magnetic Field Gradient

In the Stern-Gerlach experiment, the concept of a magnetic field gradient plays a crucial role. The gradient of a magnetic field refers to how the magnetic field strength changes over a certain distance. This change is represented as \( \frac{dB}{dz} \), where \( B \) is the magnetic field and \( z \) is the axis over which the change is measured.

A magnetic field gradient is necessary for the Stern-Gerlach experiment because it causes a force to act on particles with a magnetic moment, like the silver (Ag) atoms used in your exercise.

• This force is what ultimately causes the deflection of the particles, as they pass through the non-uniform magnetic field.
• The strength of the magnetic field gradient determines the magnitude of this force, and thus, influences the separation of the atom beams.

Understanding the concept of a magnetic field gradient is essential when working with the Stern-Gerlach experiment, as it underpins the mechanism of beam separation.

Electron Spin

Electron spin is a fundamental property of electrons that causes them to behave like tiny magnets. Think of spin as an intrinsic angular momentum that every electron possesses. This property is not due to any literal spinning of the electron, but rather is an inherent characteristic.

The notion of electrons having "up" or "down" spins corresponds to their magnetic orientation, influencing how they react in a magnetic field. In the Stern-Gerlach experiment:

• Electrons in the silver atoms are responsible for the overall magnetic moment, which interacts with the magnetic field gradient.
• When the atomic beam passes through the magnetic field gradient, electrons with different spin states divert in opposite directions. This is due to their magnetic moments interacting with the external magnetic field differently.

The concept of electron spin is integral to the understanding of atomic and molecular behavior in magnetic fields, directly affecting experimental results such as those seen in a Stern-Gerlach setup.

Magnetic Moment

The magnetic moment is a vector quantity that represents the strength and direction of a magnetic source. Any object with spin, like an electron, has an associated magnetic moment. In the Stern-Gerlach experiment, the magnetic moment is crucial as it is responsible for the force experienced by the atoms in a magnetic field gradient.

For electrons, the magnetic moment is often given in units of the Bohr magneton \( \mu_B \). This value is calculated using the formula \( \mu_B = \frac{e\hbar}{2m_e} \), where:

• \( e \) is the elementary charge,
• \( \hbar \) is the reduced Planck's constant,
• \( m_e \) is the mass of the electron.

This magnetic moment causes the atoms in the beam to deflect as they pass through the gradient, with the amount of deflection depending on the orientation of the moment relative to the magnetic field axis.

If factors change, such as the \( g \)-factor, the effective magnetic moment changes accordingly, altering the beam separation outcomes in the experiment. Understanding how the magnetic moment operates in various contexts helps clarify the Stern-Gerlach experiment's purpose of demonstrating quantized properties like spin.