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Larmor Precession Reference cheat sheet - grade 11-12

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Physics Grade 11-12

Larmor Precession Reference Cheat Sheet

A printable reference covering magnetic moments, torque, Larmor angular frequency, precession direction, gyromagnetic ratio, and resonance for grades 11-12.

TopicsMagnetismElectromagnetismTorqueRotationAtomsStudy GuidePhysics
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Larmor precession describes how a magnetic moment, such as a spinning charged particle or atomic nucleus, rotates around an external magnetic field. This reference helps students connect torque, angular momentum, and magnetic field strength in one clear model. It is useful for understanding magnetic resonance, spin behavior, and why particles do not simply line up instantly with a field.

The central idea is that a magnetic field creates a torque τ⃗=μ⃗×B⃗\vec{\tau} = \vec{\mu} \times \vec{B} on a magnetic moment. When the magnetic moment is proportional to angular momentum, μ⃗=γL⃗\vec{\mu} = \gamma \vec{L}, the torque changes the direction of L⃗\vec{L} instead of mainly changing its size. The precession angular frequency has magnitude ωL=∣γ∣B\omega_L = |\gamma|B, and the ordinary frequency is fL=ωL2πf_L = \frac{\omega_L}{2\pi}.

In quantum settings, resonance occurs when photon energy matches the spin energy splitting, hf=ΔEhf = \Delta E.

Key Facts

  • The magnetic torque on a magnetic dipole is τ⃗=μ⃗×B⃗\vec{\tau} = \vec{\mu} \times \vec{B}, so its magnitude is τ=μBsin⁡θ\tau = \mu B\sin\theta.
  • The magnetic potential energy of a dipole in a uniform magnetic field is U=−μ⃗⋅B⃗=−μBcos⁡θU = -\vec{\mu}\cdot\vec{B} = -\mu B\cos\theta.
  • If the magnetic moment is proportional to angular momentum, the relation is μ⃗=γL⃗\vec{\mu} = \gamma \vec{L}, where γ\gamma is the gyromagnetic ratio.
  • The Larmor angular frequency has magnitude ωL=∣γ∣B\omega_L = |\gamma|B, where BB is the magnetic field strength.
  • The Larmor frequency in cycles per second is fL=ωL2π=∣γ∣B2πf_L = \frac{\omega_L}{2\pi} = \frac{|\gamma|B}{2\pi}.
  • For a particle with charge qq, mass mm, and gg-factor gg, the gyromagnetic ratio is γ=gq2m\gamma = \frac{gq}{2m}.
  • The precession direction depends on the sign of γ\gamma, so positive and negative charges precess in opposite senses around B⃗\vec{B}.
  • Magnetic resonance occurs when an applied wave satisfies hf=ΔEhf = \Delta E, often matching f=fLf = f_L for spin transitions.

Vocabulary

Larmor precession
Larmor precession is the steady rotation of a magnetic moment or angular momentum vector around an external magnetic field.
Magnetic moment
A magnetic moment μ⃗\vec{\mu} measures how strongly an object behaves like a tiny magnet and how it interacts with B⃗\vec{B}.
Gyromagnetic ratio
The gyromagnetic ratio γ\gamma is the constant relating magnetic moment to angular momentum through μ⃗=γL⃗\vec{\mu} = \gamma \vec{L}.
Angular frequency
Angular frequency ω\omega measures rotational rate in radians per second, with ω=2πf\omega = 2\pi f.
Torque
Torque τ⃗\vec{\tau} is a twisting effect that changes angular momentum according to τ⃗=dL⃗dt\vec{\tau} = \frac{d\vec{L}}{dt}.
Resonance
Resonance occurs when an applied oscillating field has the correct frequency to transfer energy efficiently, such as hf=ΔEhf = \Delta E.

Common Mistakes to Avoid

  • Using fL=γBf_L = \gamma B instead of fL=∣γ∣B2πf_L = \frac{|\gamma|B}{2\pi} is wrong because γB\gamma B gives angular frequency in radians per second, not cycles per second.
  • Ignoring the sign of γ\gamma is wrong because the sign determines whether the precession is clockwise or counterclockwise around B⃗\vec{B}.
  • Treating the torque as parallel to B⃗\vec{B} is wrong because τ⃗=μ⃗×B⃗\vec{\tau} = \vec{\mu} \times \vec{B} is perpendicular to both μ⃗\vec{\mu} and B⃗\vec{B}.
  • Assuming precession always changes the size of L⃗\vec{L} is wrong because ideal Larmor precession mainly changes the direction of L⃗\vec{L} while keeping its magnitude constant.
  • Forgetting unit conversions for BB is wrong because formulas such as ωL=∣γ∣B\omega_L = |\gamma|B require magnetic field strength in teslas when using SI units.

Practice Questions

  1. 1 A proton has γ2π=42.58 MHz/T\frac{\gamma}{2\pi} = 42.58\,\text{MHz/T}. What is its Larmor frequency in a field of B=1.50 TB = 1.50\,\text{T}?
  2. 2 An electron spin has ∣γ∣=1.76×1011 rad s−1T−1|\gamma| = 1.76 \times 10^{11}\,\text{rad}\,\text{s}^{-1}\text{T}^{-1}. Find ωL\omega_L when B=0.020 TB = 0.020\,\text{T}.
  3. 3 A magnetic moment has μ=3.0×10−23 J/T\mu = 3.0 \times 10^{-23}\,\text{J/T} in a field B=0.40 TB = 0.40\,\text{T} at an angle of 30∘30^{\circ}. Calculate the torque magnitude using τ=μBsin⁡θ\tau = \mu B\sin\theta.
  4. 4 Why does a magnetic moment precess around B⃗\vec{B} instead of immediately pointing exactly along B⃗\vec{B} when angular momentum is present?

Understanding Larmor Precession Reference

A useful way to picture precession is to think of a spinning top in gravity. Gravity pulls downward, but the top's spin makes its axis sweep around instead of falling straight down. A magnetic moment behaves similarly in a magnetic field.

The turning effect acts sideways relative to the moment and the field. Because this turning effect is sideways, it continually redirects the angular momentum. It does not usually make the angular momentum grow or shrink.

The angle between the spin direction and the field can therefore remain nearly constant while the direction traces out a cone. This cone picture is one of the most important ideas to draw when studying precession.

The rate of this motion depends on two physical features. A stronger external field produces a faster sweep around the field direction. A larger gyromagnetic ratio also produces a faster sweep.

The gyromagnetic ratio connects magnetic behavior to angular momentum for a particular particle or nucleus. It is not the same for every object. Electrons, protons, and different atomic nuclei respond at different rates in the same field.

The sign matters as much as the size in direction questions. A positive gyromagnetic ratio gives one sense of rotation, while a negative value gives the opposite sense. Use a right hand rule only after checking which vector relation and which sign convention the problem gives.

Energy helps explain why alignment is favored without guaranteeing immediate alignment. A moment pointing in the energetically preferred direction has lower magnetic potential energy than one pointing against the field. Yet a perfectly isolated magnetic system can keep precessing instead of settling down.

To lose energy and move toward the lowest energy state, it needs some interaction with its surroundings. In real materials, nearby atoms, molecular motion, and electromagnetic effects can transfer energy away. This process is called relaxation in magnetic resonance.

Precession describes the motion around the field. Relaxation describes the gradual change in the motion and population of energy states caused by the environment. Keeping these ideas separate prevents a common mistake.

Magnetic resonance uses a carefully chosen alternating magnetic field to affect precessing moments. The applied field can transfer energy efficiently when its frequency matches the natural precession frequency. This is similar to pushing a swing at the right rhythm.

In a quantum description, only certain spin energy states are allowed. Absorption occurs when the incoming radiation supplies exactly the energy gap between states. In medical MRI, hydrogen nuclei in water and fat are placed in a strong field, excited by radio waves, then measured as they relax.

Chemists use nuclear magnetic resonance to identify molecular structures. In calculations, watch the units closely. Angular frequency is measured in radians per second, while ordinary frequency is measured in cycles per second.

They differ by a factor of two times pi. Also distinguish a uniform field, which mainly causes steady precession, from a nonuniform field, which can exert a net force and spread out the precession rates.