Force on a Magnetic Dipole in an Inhomogeneous Magnetic Field
Conditions: Magnetic dipole placed in a magnetic field pointing predominantly along the z-axis with a spatial gradient along z.
Result:
Modern Physics
Stern-Gerlach Experiment
The Stern-Gerlach experiment demonstrated the spatial quantization of angular momentum by passing a beam of silver atoms through an inhomogeneous magnetic field. It provided direct experimental evidence for electron spin and space quantization.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of the deflection of atomic beams, spatial quantization, and the concept of electron spin with two possible orientations.
PGT extension
Detailed mathematical derivation of the magnetic force on a magnetic dipole, calculation of splitting in inhomogeneous fields, Hamiltonian of the interaction, and Larmor precession.
Prerequisites
- Bohr's theory of hydrogen atom
- Electron spin and magnetic dipole moment
- Spatial quantization
Historical Context and Setup
- Conducted by Otto Stern and Walther Gerlach in 1922 in Frankfurt, Germany.
- Utilized an atomic beam of silver (Ag) atoms evaporated from an electrically heated oven in a high vacuum.
- The beam passed through a specially shaped inhomogeneous magnetic field created by an asymmetric pole piece.
Role of Inhomogeneous Magnetic Field
- A uniform magnetic field exerts a torque on a magnetic dipole but no net translational force.
- An inhomogeneous magnetic field exerts a net translational force proportional to the spatial derivative of the field component along the dipole axis.
- The force is given by $F_z = \mu_z (d B_z / d z)$, which splits the beam into distinct spatial traces.
Observation and Spatial Quantization
- Classical physics predicts a continuous smearing or a broad single band on the detector plate corresponding to random orientations of magnetic moments.
- The experiment observed the beam splitting into two discrete traces, proving that the magnetic moment component along the field can only take specific quantized values.
- For silver atoms, the ground state has total orbital angular momentum L = 0, meaning the observed splitting is entirely due to the intrinsic spin angular momentum ($S = 1/2) of the valence electron$.
Implications for Quantum Mechanics
- Confirmed the space quantization rule where the z-component of angular momentum is m_s hbar, with $m_s = +/- 1/2$ for an electron.
- Laid the foundation for modern magnetic resonance techniques and established spin as an inherent quantum degree of freedom.
- Directly influenced the formulation of Pauli's exclusion principle and matrix mechanics.
Derivations
Deflection of Atomic Beam in Stern-Gerlach Apparatus
Conditions: Silver atoms of mass m and velocity v passing through a magnetic field of length L with gradient d B_z / d z.
- Write the potential energy of the magnetic dipole in the field: $U = - \mu \cdot B = - \mu_z B_z$.
- Determine the force along the z-direction: $F_z = - (d U / d z) = \mu_z (d B_z / d z)$.
- Calculate the acceleration experienced by the atom: $a_z = F_z / M = \frac{\mu_z}{M} \frac{d B_z}{d z}$.
- Determine the time spent in the magnetic field of length L: $t = L / v$.
- Calculate the spatial deflection d along the z-axis on the detector: $d = \frac{1}{2} a_z t^2 = \frac{1}{2} \frac{\mu_z}{M} \frac{d B_z}{d z} \left(\frac{L}{v}\right)^2$.
Result: Deflection $d = \frac{1}{2 M} \mu_z \left(\frac{d B_z}{d z}\right) \left(\frac{L}{v}\right)^2$
Formula sheet
| Formula / law | Equation | Variables and units | Conditions |
|---|---|---|---|
| Magnetic Moment of Electron Spin | $$\mu_s = - g_s \frac{e}{2 m_e} S$$ | m$u_s = spin magnetic moment$, $g_s = gyromagnetic ratio (~2)$, e = electronic charge, $m_e = electron mass$, S = spin angular momentum A m^2 or J T^-1 | Non-relativistic quantum mechanics |
| Z-component of Spin Magnetic Moment | $$\mu_{sz} = \pm \mu_B$$ | m$u_sz = z-component of spin magnetic moment$, m$u_B = Bohr magneton$ J T^-1 | For a spin-1/2 particle |
| Bohr Magneton | $$\mu_B = \frac{e \hbar}{2 m_e}$$ | e = elementary charge, hbar = reduced Planck's constant, $m_e = mass of electron$ J T^-1 | Fundamental atomic constant |
| Force on Magnetic Dipole | $$F_z = \mu_z \frac{d B_z}{d z}$$ | $F_z = force in z direction$, m$u_z = magnetic moment component$, d B_z / d z = magnetic field gradient Newtons (N) | Inhomogeneous magnetic field |
| Spatial Quantization Condition | $$J_z = m_j \hbar$$ | $J_z = z-component of total angular momentum$, $m_j = magnetic quantum number$ J s | Any quantized angular momentum system |
| Potential Energy in Magnetic Field | $$U = - \mu_z B_z$$ | U = potential energy, m$u_z = magnetic moment component$, $B_z = magnetic field$ Joules (J) | Alignment along z-axis |
| Number of Beam Components | $$N = 2 j + 1$$ | N = number of split components, j = total angular momentum quantum number Dimensionless integer | Space quantization rule for half-integer or integer spin |
| Deflection Distance | $$z = \frac{\mu_z L^2}{2 m v^2} \frac{d B_z}{d z}$$ | z = vertical deflection, L = length of field, m = atomic mass, v = velocity Meters (m) | Constant magnetic field gradient |
Graphs and experiments
Stern-Gerlach Experimental Setup
Draw: An oven emitting silver atoms, a collimating slit, an inhomogeneous magnet with wedge-shaped and flat pole pieces, and a detector plate showing two split traces.
Interpret: Illustrates how the atomic beam splits symmetrically into two discrete beams corresponding to $m_s = +1/2$ and $m_s = -1/2$.
Exam use: Frequently asked in descriptive questions to identify components and explain the necessity of the inhomogeneous field.
Detector Plate Pattern (Classical vs. Quantum)
Draw: Comparison of expected continuous vertical smear (classical prediction) versus two distinct parallel lines (quantum observation).
Interpret: Contrasts classical physics expectation of continuous orientation with quantum mechanics reality of discrete spatial quantization.
Exam use: Tested in objective questions to distinguish between classical and quantum mechanical outcomes.
Sequential Stern-Gerlach Filters
Draw: Three apparatuses in series: SG_z (selects +z), SG_x (selects +x), and a second SG_z.
Interpret: Demonstrates quantum measurement collapse, showing that measuring spin along x destroys the previously determined z-component information.
Exam use: Advanced conceptual PGT questions on quantum state superposition and measurement.
Solved numericals
A beam of silver atoms passes through a Stern-Gerlach apparatus. How many traces are observed on the detector plate, and what does this indicate about the total angular momentum of the silver atom in its ground state?
- Recall the formula for the number of split components: $N = 2j + 1$.
- For silver atoms in the ground state, the valence electron is in the 5s orbital, so orbital angular momentum L = 0 and total spin $S = 1/2$.
- Therefore, total angular momentum $j = 1/2$.
- Calculate N: $N = 2(1/2) + 1 = 2$.
Answer: 2 traces; indicates total angular momentum $j = 1/2 (due$ to electron spin).
Calculate the force acting on a silver atom in a Stern-Gerlach magnet if the magnetic field gradient (d B_z / d z) is 1.5 x 1-^1 T/m. Assume the magnetic moment of the silver atom is equal to one Bohr magneton (m$u_B = 9.27 x 10^-24 J/T)$.
- Identify the formula for force on a magnetic dipole: $F_z = \mu_z (d B_z / d z)$.
- Substitute the value of the magnetic moment m$u_z = \mu_B = 9.27 x 10^-24 J/T$.
- Substitute the field gradient d B_z / d $z = 1.5 x 1^1 T/m$.
- Compute the product: $F_z = (9.27 x 10^-24) x (1.5 x 1^1) = 1.39 x 10^-11 N$.
Answer: 1.39 x 10^-11 N
Find the vertical deflection of a silver atom (mass $m = 1.79 x 10^-25 kg) passing through an inhomogeneous magnetic field of length L = 5.0 cm with a gradient of 1.0 x 1^2 T/m$. Assume the atomic velocity is 500 m/s and m$u_z = 9.27 x 10^-24 J/T$.
- Use the deflection formula: $z = \frac{\mu_z L^2}{2 m v^2} \frac{d B_z}{d z}$.
- Substitute the given values: m$u_z = 9.27 x 10^-24 J/T$, L = 0.05 m, $m = 1.79 x 10^-25 kg$, $v = 500 m/s$, d B_z / d $z = 1.0 x 1^2 T/m$.
- Calculate $L^2 = (0.05)^2 = 2.5 x 10^-3 m^2$.
- Calculate $v^2 = (500)^2 = 2.5 x 10^5 m^2/s^2$.
- Evaluate the expression: $z = \frac{9.27 \times 10^{-24} \times 2.5 \times 10^{-3} \times 1.0 \times 10^2}{2 \times 1.79 \times 10^{-25} \times 2.5 \times 10^5}$.
- Simplify numerator: 2.3175 x 10^-24.
- Simplify denominator: 8.95 x 10^-20.
- Final division: z approx 2.59 x 10^-5 m.
Answer: 2.59 x 10^-5 m (or 0.0259 mm)
Assumptions and limitations
- Assumes the atomic beam is well-collimated and atoms do not interact with one another inside the beam (ideal gas approximation).
- Neglects relativistic corrections to electron spin magnetic moment.
- Valid only when the external magnetic field gradient is strong enough to overcome internal coupling effects (Paschen-Back regime relative to fine structure).
Common misconceptions
Common error: The Stern-Gerlach experiment proves the orbital angular momentum of electrons is quantized.
Correct: In silver atoms, the orbital angular momentum is zero (L = 0), so the experiment specifically proves the quantization of electron spin.
Silver ground state has a single valence electron in an s-orbital, possessing no orbital angular momentum.
Common error: A uniform magnetic field is required to split the atomic beam.
Correct: An inhomogeneous (non-uniform) magnetic field is strictly necessary to exert a translational force on the magnetic dipole.
A uniform field exerts only a torque, causing precession without translational deflection.
Common error: Classical mechanics can explain the two discrete spots on the detector.
Correct: Classical physics predicts a continuous continuous band due to random dipole orientations.
Discrete splitting is a purely quantum mechanical manifestation of spatial quantization.
Quick revision
- The Stern-Gerlach experiment verified space quantization and electron spin using a beam of silver atoms in an inhomogeneous magnetic field.
- Magnetic Moment of Electron Spin: \m$u_s = - g_s \frac{e}{2 m_e} S$
- Z-component of Spin Magnetic Moment: \m$u_{sz} = \pm \mu_B$
Must remember
- The Stern-Gerlach experiment was conducted in 1922 by Otto Stern and Walther Gerlach using a beam of silver (Ag) atoms.
- An inhomogeneous (non-uniform) magnetic field is strictly required to exert a net translational force on magnetic dipoles.
- A uniform magnetic field only exerts a torque and causes precession, resulting in no translational deflection or beam splitting.
- Silver atoms in the ground state have total orbital angular momentum $L = 0$, meaning the observed splitting into two components is due entirely to electron spin ($S = 1/2$).
- The number of split components for a given angular momentum quantum number $j$ is given by $N = 2j + 1$.
- The force on a magnetic dipole in an inhomogeneous field along the z-axis is $F_z = \mu_z \frac{d B_z}{d z}$.
- Classical physics incorrectly predicts a continuous smearing or broad single band instead of discrete traces.
- The Bohr magneton $\mu_B = \frac{e \hbar}{2 m_e}$ serves as the natural unit for expressing electron magnetic moments.
Glossary
Inhomogeneous Magnetic FieldA magnetic field that varies in space, possessing a non-zero spatial derivative (gradient), necessary to exert a translational force on a magnetic dipole.
Spatial QuantizationThe quantum mechanical phenomenon where the direction of the angular momentum vector can only take specific quantized orientations relative to a reference axis.
Bohr Magneton ($\mu_B$)A physical constant and natural unit for expressing the magnetic dipole moment of an electron caused by its orbital or spin angular momentum.
Silver Atom (Ag)The atom used in the original Stern-Gerlach experiment, chosen because its single valence electron in an s-orbital ($L=0$) isolates the spin magnetic moment.
Gyromagnetic Ratio ($g_s$)The ratio of the magnetic dipole moment to the angular momentum; for electron spin, $g_s \approx 2$.
Translational ForceA force that causes linear acceleration of a particle, generated in the Stern-Gerlach setup by the interaction between a magnetic dipole and a field gradient.
Uniform vs. Inhomogeneous Magnetic Fields
| Feature | Uniform Magnetic Field | Inhomogeneous Magnetic Field |
|---|---|---|
| Spatial Variation | Constant magnitude and direction ($\frac{d B}{d z} = 0$) | Varies in space with a non-zero gradient ($\frac{d B}{d z} \neq 0$) |
| Force on Magnetic Dipole | Zero net translational force ($F = 0$) | Non-zero net translational force ($F_z = \mu_z \frac{d B_z}{d z}$) |
| Effect on Beam | Exerts torque, causes precession without spatial splitting | Splits the atomic beam into discrete traces corresponding to $m_s$ |
Classical vs. Quantum Predictions
| Aspect | Classical Prediction | Quantum Mechanical Prediction |
|---|---|---|
| Beam Trace | Continuous smearing or single broad band | Discrete, separated traces ($N = 2j + 1$) |
| Orientation of Dipoles | Random, continuous distribution of angles | Quantized orientations relative to the magnetic field axis |
| Underlying Principle | Continuous energy and momentum states | Space quantization and intrinsic spin |
Practice Quiz
Choose an option to reveal the explanation.
1. Who conducted the historic experiment demonstrating space quantization in 1922?
Explanation: The Stern-Gerlach experiment was successfully conducted by Otto Stern and Walther Gerlach in Frankfurt, Germany, in 1922.
2. What type of magnetic field is required to split an atomic beam in the Stern-Gerlach experiment?
Explanation: An inhomogeneous magnetic field is strictly required to exert a net translational force ($F_z = \mu_z \frac{d B_z}{d z}$) on a magnetic dipole.
3. Why were silver (Ag) atoms used in the original Stern-Gerlach experiment?
Explanation: Silver atoms have a single valence electron in an s-orbital, meaning orbital angular momentum $L = 0$, so the observed splitting is entirely due to spin ($S = 1/2$).
4. What is the force experienced by a magnetic dipole with z-component $\mu_z$ in a magnetic field gradient $\frac{d B_z}{d z}$?
Explanation: The translational force in an inhomogeneous magnetic field is given by $F_z = \mu_z \frac{d B_z}{d z}$. ज़ाहिर है कि यह फील्ड ग्रेडिएंट पर निर्भर करता है।
5. How many split components ($N$) does a beam of particles with total angular momentum quantum number $j$ produce?
Explanation: According to the space quantization rule, the number of split components is $N = 2j + 1$. For $j = 1/2$, this yields $2(1/2) + 1 = 2$ components.
6. What does classical physics predict for the beam pattern on the detector plate in the Stern-Gerlach setup?
Explanation: Classical physics predicts that magnetic moments can take any random orientation, leading to a continuous smearing or broad single band on the detector plate.
7. What is the expression for the Bohr magneton ($\mu_B$)?
Explanation: The fundamental atomic constant Bohr magneton is defined as $\mu_B = \frac{e \hbar}{2 m_e}$.
8. What happens when a magnetic dipole is placed in a purely uniform magnetic field?
Explanation: A uniform magnetic field exerts a torque on a magnetic dipole causing Larmor precession, but no net translational force.
9. What is the z-component of the spin magnetic moment for an electron ($s = 1/2$)?
Explanation: For an electron with spin-1/2, the z-component of the spin magnetic moment is quantized as $\mu_{sz} = \pm \mu_B$.
10. Which quantum property of the electron was directly confirmed by the Stern-Gerlach experiment?
Explanation: Since silver has $L = 0$, the two-way splitting directly confirmed the intrinsic spin angular momentum and its spatial quantization ($m_s = \pm 1/2$).
11. What is the formula for the potential energy $U$ of a magnetic dipole aligned along a magnetic field $B_z$?
Explanation: The potential energy of a magnetic dipole in a magnetic field is given by $U = - \vec{\mu} \cdot \vec{B}$, which for the z-axis is $U = - \mu_z B_z$.
12. What is the approximate value of the gyromagnetic ratio $g_s$ for electron spin?
Explanation: The gyromagnetic ratio for electron spin $g_s$ is approximately 2 (and slightly more precisely $2.0023$ due to quantum electrodynamics).
13. Which of the following is a common misconception regarding the Stern-Gerlach experiment?
Explanation: Silver atoms have $L = 0$ (orbital angular momentum is zero), so it does not prove orbital angular momentum quantization; it proves spin quantization.
14. What units are used for the magnetic moment $\mu_s$?
Explanation: Magnetic dipole moment units are Joules per Tesla ($\text{J T}^{-1}$) or Ampere-square meters ($\text{A m}^2$).
15. What does the deflection distance $z$ of the atomic beam depend upon in a constant magnetic field gradient?
Explanation: From the deflection formula $z = \frac{\mu_z L^2}{2 m v^2} \frac{d B_z}{d z}$, it depends on $\mu_z$, field gradient, length $L$, mass $m$, and velocity $v$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A beam of silver atoms is passed through an inhomogeneous magnetic field in a Stern-Gerlach apparatus. If the beam splits into two traces, what does this primarily indicate about the silver atoms?
Explanation: Splitting into two traces ($N = 2j + 1 = 2 \implies j = 1/2$) for silver ($L=0$) indicates the valence electron has spin $S = 1/2$.
2. Why is a uniform magnetic field inadequate for observing beam splitting in the Stern-Gerlach experiment?
Explanation: In a uniform field, the forces on the north and south poles of the dipole cancel out, yielding zero net translational force (though a torque exists).
3. If a particle with total angular momentum quantum number $j = 3/2$ is passed through a Stern-Gerlach apparatus with an inhomogeneous magnetic field, into how many components will the beam split?
Explanation: Using the space quantization formula $N = 2j + 1$, for $j = 3/2$, we get $N = 2(3/2) + 1 = 4$ components.
4. In the expression for the force on a magnetic dipole $F_z = \mu_z \frac{d B_z}{d z}$, what physical condition must be met for $F_z$ to be non-zero?
Explanation: A non-zero spatial derivative or gradient ($\frac{d B_z}{d z} \neq 0$) is required to produce a translational force.
5. What is the physical significance of the Bohr magneton $\mu_B$ in atomic physics?
Explanation: The Bohr magneton $\mu_B = \frac{e \hbar}{2 m_e}$ is the fundamental natural unit for electron magnetic dipole moments.
6. According to classical electromagnetic theory, what should be observed on the detector plate in a Stern-Gerlach experiment?
Explanation: Classically, magnetic dipoles can take any arbitrary angle relative to the field, producing a continuous band rather than discrete spots.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In a Stern-Gerlach experiment using silver atoms, what is the total orbital angular momentum quantum number $L$ of the ground state?
Explanation: Ground state silver atoms have their valence electron in a $5s$ orbital, which has an orbital angular momentum quantum number $L = 0$.
2. Which equation correctly gives the z-component of the spin angular momentum for an electron?
Explanation: The z-component of spin angular momentum is quantized as $S_z = m_s \hbar$, where $m_s = \pm 1/2$ for an electron.
3. If the magnetic field gradient $\frac{d B_z}{d z}$ is doubled in a Stern-Gerlach setup while keeping all other parameters constant, what happens to the vertical deflection $z$?
Explanation: Deflection $z$ is directly proportional to the magnetic field gradient ($\frac{d B_z}{d z}$), so doubling the gradient doubles the deflection.
4. What physical parameter does NOT affect the spatial splitting in the Stern-Gerlach detector plate?
Explanation: The primary magnetic deflection formula $z = \frac{\mu_z L^2}{2 m v^2} \frac{d B_z}{d z}$ depends on velocity, length, mass, and gradient, while standard gravity is negligible compared to magnetic forces.
5. The potential energy $U$ of a magnetic dipole in an inhomogeneous magnetic field is minimum when:
Explanation: Since $U = - \mu_z B_z$, energy is minimum when $\mu_z$ and $B_z$ are aligned in the same direction (parallel, $\theta = 0^{\circ}$).
6. What is the main reason why atoms in the Stern-Gerlach apparatus are produced via an electrically heated oven in a high vacuum?
Explanation: A high vacuum ensures that the evaporated atoms travel in a well-defined beam without scattering off gas molecules.
7. Which principle of quantum mechanics is most directly demonstrated by the Stern-Gerlach experiment?
Explanation: The experiment directly demonstrates space quantization, showing that angular momentum components can only take discrete quantized values along a field axis.
8. Calculate the number of beam components expected for an atomic beam with total angular momentum $j = 1$.
Explanation: Using $N = 2j + 1$, for $j = 1$, we get $N = 2(1) + 1 = 3$ components ($m_j = 1, 0, -1$).
9. How does the magnetic moment $\mu_s$ relate to the spin angular momentum $S$ in non-relativistic quantum mechanics?
Explanation: The magnetic moment due to spin is given by $\mu_s = - g_s \frac{e}{2 m_e} S$, incorporating the gyromagnetic ratio and negative charge of the electron.
10. If a particle beam passes through successive Stern-Gerlach analyzers (e.g., $SG_z$ followed by $SG_x$ and another $SG_z$), what fundamental quantum mechanical feature is illustrated?
Explanation: Sequential Stern-Gerlach measurements illustrate that measurements along non-commuting axes (like z and x) interfere with and alter the previously prepared quantum states due to wave function collapse.