(II) In a Stern-Gerlach experiment, Ag atoms exit the oven...

(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?

Key Concepts

Stern-Gerlach Experiment

This experiment demonstrates the quantization of angular momentum by sending an atomic beam through a non-uniform magnetic field, which causes the atoms to deflect according to their magnetic moments. The spatial separation of the beam into discrete components provides direct evidence of the quantized nature of intrinsic spin and magnetic dipoles in atoms.

Magnetic Dipole Interaction

Magnetic dipoles, such as those associated with atomic magnetic moments, experience a force in a non-uniform magnetic field that is proportional to the gradient of the field and the component of the dipole moment along the field direction. This interaction is responsible for the deflection of atoms in the Stern-Gerlach experiment, leading to measurable separations between different quantum states.

g-factor

The g-factor is a dimensionless constant that relates the magnetic moment of a particle to its angular momentum. In the context of the Stern-Gerlach experiment, the value of the g-factor determines the magnitude of the magnetic moment and, consequently, the extent of the deflection in a magnetic field gradient. Variations in the g-factor can lead to different separations between deflected beams, reflecting differences in intrinsic spin properties.

Atomic Beam Dynamics

Understanding the motion of the atomic beam involves applying principles of kinematics, including the calculation of transverse acceleration due to magnetic forces and the resulting deflection over a given distance. These dynamics integrate the effects of the initial velocity of atoms, the strength and gradient of the magnetic field, and the interaction time within the magnetic region, all critical for predicting the beam separation observed in experiments.

Transcript

00:01 The atom will go through a uniformly accelerated movement because of the influence of the magnetic field.

00:09 Then, its deviation from the horizontal is given by this very simple and well -known equation of motion.

00:16 Y of t is equal to y0 plus v0 times t plus the acceleration times times squared divided by 2.

00:31 In this question, we suppose that it initially only had velocity on the x direction.

00:39 So, v0 for the y direction is equal to zero.

00:43 And we also suppose that they all begin their movement precisely at the same horizontal line, which is centered at y0 equals to zero.

00:53 So that deviation from the horizontal is simple given by one half of the acceleration, times the time squared.

01:04 Now to calculate the acceleration we can use newton's second law which says that f is equal to the mass times the acceleration.

01:15 Then the deviation is now given by one half times the force divided by the mass of that atom times squared.

01:31 So our next task is then calculate what is the force.

01:36 The force is due to a magnetic field.

01:38 Then it's just given by minus g mubi times b, sorry, not b, times db db, x times the spin quantum number.

01:59 So our deviation is equal to 1 half of minus g, times mubi times d b d z divided by m a g times the spin quantum number now we have to evaluate we have to know what is the amount of time it takes to travel from the beginning to the end of the stern galarac apparatus knows that we have x velocity of 780 meters by second and an horizontal distance of only 5 centimeters...

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