A charged sphere sits stationary inside a solenoid carrying a steady current. Before the solenoid is switched off, what is the angular momentum of the system?
AZero — nothing is moving, so no angular momentum is present
BNonzero — angular momentum is stored in the electromagnetic field configuration, even with static fields and no motion
CNonzero — the charged sphere is slowly precessing due to the static magnetic field
DUndefined — angular momentum requires a well-defined rotation axis, and this system is symmetric
This is the core lesson of electromagnetic angular momentum: the fields themselves carry angular momentum via l = ε₀(r × (E × B)). The electric field (from the charge) and magnetic field (from the solenoid) together store angular momentum throughout the overlapping field region, even though neither object is moving. When the solenoid is switched off, the disappearing B field induces an E field (Faraday's law) that exerts a tangential force on the charged sphere, setting it rotating — field angular momentum converts to mechanical angular momentum. This is the Feynman disk paradox.
Question 2 Multiple Choice
Orbital angular momentum (OAM) and spin angular momentum (SAM) of a light beam differ in which of the following ways?
AOAM arises from circular polarization; SAM arises from helical phase structure in the wavefront
BOAM arises from helical phase structure in the wavefront; SAM arises from circular or elliptical polarization
COAM and SAM are two names for the same physical quantity — the total angular momentum of the beam
DOAM is purely quantum mechanical, while SAM is a classical wave property
The two types are physically distinct and arise from different aspects of the field. Spin angular momentum (SAM) is tied to polarization state: circularly polarized light carries ±ℏ per photon. Orbital angular momentum (OAM) arises from helical phase structure — the phase winds by 2πℓ around the beam axis, carrying ℓℏ per photon. They can be manipulated independently using optical elements (waveplates change SAM; spiral phase plates change OAM), which has enabled independent control in optical tweezer and communications applications.
Question 3 True / False
When the current in a solenoid enclosing a stationary charged sphere is switched off, the sphere begins to rotate — and this rotation is explained by conservation of angular momentum.
TTrue
FFalse
Answer: True
This is exactly the Feynman disk paradox. The changing B field induces a tangential E field (Faraday's law) that exerts a torque on the charged sphere. The sphere acquires mechanical angular momentum precisely equal to the angular momentum that was stored in the crossed E and B fields. Total angular momentum is conserved: field angular momentum converts to mechanical angular momentum. Nothing was 'spinning' before, yet the system had nonzero angular momentum stored in the fields.
Question 4 True / False
The electromagnetic angular momentum density at a point requires a moving charge at that point — it cannot be nonzero in the empty space between a static charge distribution and a static magnetic field.
TTrue
FFalse
Answer: False
The angular momentum density l = ε₀(r × (E × B)) depends only on the values of E and B at each point — not on the presence of charges there. Wherever both an electric field and a magnetic field are present with appropriate spatial geometry, angular momentum is stored in the fields even in vacuum. The charged sphere + solenoid example is precisely a static configuration with angular momentum distributed throughout the space where both E and B overlap. Fields carry energy, momentum, and angular momentum as properties of space, not just of charges.
Question 5 Short Answer
The Feynman disk paradox involves a stationary charged ring and a solenoid with no moving parts. How does this demonstrate that angular momentum can be stored in electromagnetic fields, and what happens when the solenoid is switched off?
Think about your answer, then reveal below.
Model answer: The charged ring produces a radial electric field E; the solenoid produces an axial magnetic field B. The angular momentum density l = ε₀(r × (E × B)) is nonzero throughout the space where both fields overlap, even though nothing is moving. Initial mechanical angular momentum is zero, but total angular momentum (field + mechanical) is nonzero. When the solenoid is switched off, the disappearing B field induces an azimuthal E field by Faraday's law, which exerts a tangential force on the charged ring, causing it to rotate. The mechanical angular momentum gained by the ring equals the field angular momentum that was initially stored — angular momentum is conserved, converted from field form to mechanical form.
The paradox resolves only if you accept that fields are real physical entities carrying conserved quantities, not just calculational conveniences. The pre-switch state has zero mechanical angular momentum but nonzero total angular momentum; the post-switch state has that angular momentum carried mechanically. This makes field angular momentum physically real in the strongest sense: it participates in conservation laws.