Difficulty: Intermediate | Prerequisites: Coulomb's Law, electric field lines, vector components, Newton's Second Law, concept of potential energy from Physics 211
Once you can calculate electric forces and fields, the natural next question is: can you balance them against other forces to hold a charge in place? This week's problems explore electrostatic equilibrium, where electric and gravitational forces cancel, and ask whether that balance is stable (the charge returns when nudged) or unstable (the charge flies away). The quadrupole problem then shows how arranged charge configurations can shape and steer beams of charged particles, a technique used in real accelerators and old television sets. Together these problems connect abstract force calculations to physical intuition about what charges actually do in three dimensions.
A charged particle can be suspended in mid-air by balancing Coulomb repulsion against gravity. That equilibrium is stable in some directions but unstable in others, which is a fundamental consequence of how electric fields behave in three dimensions. Quadrupole arrangements of charges focus beams in one axis while defocusing in another, and pairing two quadrupoles achieves net focusing in both.
Electrostatic equilibrium
A condition in which the net force on a charged particle is zero because electric forces exactly cancel other forces (here, gravity). In simple terms, the charge just floats because the upward electrical push matches the downward gravitational pull.
Stable equilibrium
An equilibrium where a small displacement produces a restoring force that pushes the object back to its resting position. Think of a marble sitting at the bottom of a bowl: nudge it and it rolls back.
Unstable equilibrium
An equilibrium where a small displacement produces a force that drives the object further away from its resting position. Think of a marble balanced on top of a hill: the slightest bump sends it rolling off.
Restoring force
A force that acts opposite to the direction of displacement, pulling the object back toward equilibrium. In simple terms, if you push it right, the force pushes it left.
Limiting case (limiting behaviour)
A check on a derived formula by examining what happens when a parameter goes to zero or infinity. If the formula gives a physically sensible result in these extremes, you gain confidence it is correct.
Electrostatic quadrupole
An arrangement of four parallel charged rails (two positive, two negative) positioned symmetrically. The resulting field focuses charged particles along one axis while defocusing them along the perpendicular axis.
Linear charge density (λ)
Charge per unit length along a rod or rail, measured in C/m. Think of it as how densely packed the charge is along the length of the wire.
Two fixed charges Q are placed a horizontal distance d apart.
A third charge q (mass m) is placed at height h above the midpoint of the two fixed charges.
For equilibrium, the vertical components of the two Coulomb forces must exactly cancel gravity:
2 × [kQq / ((d/2)² + h²)] × [h / √((d/2)² + h²)] = mg
The horizontal components cancel by symmetry (equal and opposite).
Solving for q:
q = (mg / 2kQh) × [(d/2)² + h²]^(3/2)
The sign of q must match the sign of Q so that the electric force is repulsive (pushing q upward).
Limiting-case analysis is a powerful habit. Three checks for this problem:
As h → 0: q → ∞. This makes sense. If the charge is right between the two fixed charges at ground level, the vertical component of the electric force vanishes, so you would need an infinitely large charge to generate any upward force at all.
As d → 0: The two fixed charges sit on top of each other and act like a single charge 2Q directly below q. The formula reduces to q = mgh² / (2kQ), which is the simple one-dimensional Coulomb balance you would expect.
As m → 0: q → 0. A massless particle needs no electric force to float, so zero charge is correct.
The equilibrium is not equally robust in every direction. You must test x, y, and z independently.
Displacement in y (vertical):
If q moves up, the electric force weakens (greater distance from Q charges) while gravity stays constant, so the net force pulls q back down.
If q moves down, the electric force strengthens and pushes q back up.
Result: stable in y. The force-versus-position plot shows a negative slope through the equilibrium point (force opposes displacement).
Displacement in x (horizontal, along the line joining the two Q charges):
If q shifts left or right, the symmetry of the two Q charges produces a net horizontal force pointing back toward the midpoint.
Result: stable in x. Same restoring-force logic as the y direction.
Displacement in z (horizontal, perpendicular to the line joining the two Q charges, i.e. in and out of the page):
If q shifts in the z direction, both Q charges are now farther away than before (the perpendicular distance increases), and there is no component of the Coulomb force pulling q back toward z = 0.
The force on either side of z = 0 points in the same direction as the displacement.
Result: unstable in z. Any small perturbation in z sends q drifting further away.
Overall verdict: The equilibrium is unstable, because stability requires restoring forces in all three directions, and z fails.
A system in stable equilibrium sits at a local minimum of potential energy (a "well" or "bowl" shape).
A system in unstable equilibrium sits at a local maximum or saddle point.
The potential energy surface U(x, y, z) for this problem shows a bowl in the x-y plane but a ridge along z, confirming the saddle-point nature of the equilibrium.
This connects to the broader physics principle that electrostatic equilibrium for a free charge in empty space is always unstable in at least one direction (a consequence of Earnshaw's theorem, which you may meet later in the course).
Four long parallel rails form a square cross-section of side d:
Left and right rails: positive charge density +λ.
Top and bottom rails: negative charge density −λ.
Field lines run from the positive rails toward the negative rails, curving through the interior.
Field strength is greatest near any rail (pole) and weakest at the centre of the arrangement.
Effect on an electron beam:
Electrons entering along the axis of the quadrupole (parallel to the rails) experience transverse forces from the field.
The field pushes electrons inward along one transverse axis (say x) and outward along the perpendicular axis (say y).
A circular beam becomes elliptical: compressed in y, stretched in x.
Why use pairs of quadrupoles:
A single quadrupole focuses in one plane and defocuses in the other.
Rotating a second quadrupole by 90° and placing it downstream reverses the focusing and defocusing axes.
The net effect of the pair is focusing in both transverse directions (a principle called "strong focusing" or "alternating-gradient focusing").
CRT television sets used this to correct imperfections in the electron beam and ensure it hit the correct pixel on the screen.
Quantity | Expression | Notes |
|---|---|---|
Equilibrium charge | q = (mg / 2kQh) × [(d/2)² + h²]^(3/2) | Balances gravity against vertical Coulomb components |
Limiting case d → 0 | q = mgh² / (2kQ) | Reduces to single-source balance |
Force in uniform field | F = qE | Always applies once you know E |
Quadrupole magnets (the magnetic analogue of the electrostatic quadrupole described here) are used in every major particle accelerator in the world, including the Large Hadron Collider at CERN. They keep proton beams tightly focused as they circulate at near light speed. The same alternating-gradient focusing principle was also essential in CRT monitors and television tubes, where pairs of quadrupole deflectors steered the electron beam to the correct spot on the phosphor screen.
Earnshaw's theorem (the impossibility of stable electrostatic equilibrium in free space) is the reason magnetic traps and Paul traps use oscillating fields rather than static ones to confine charged particles, a technique central to quantum computing experiments with trapped ions.
"If the charge is in equilibrium, it must be stable." Equilibrium only means the net force is zero at that point. Stability requires that small displacements produce restoring forces in every direction. This problem shows that you can have stability in two directions and instability in a third.
"Checking limiting cases is optional." It is one of the most reliable ways to catch algebraic errors. If your formula gives q → 0 when m → ∞, something is wrong, and you will find it before the exam does.
"A quadrupole focuses a beam." A single quadrupole focuses in one plane and defocuses in the other. You need a pair, rotated 90° relative to each other, to achieve net focusing in both transverse directions.
"The equilibrium charge q should have the opposite sign from Q." If q and Q have opposite signs, the electric force is attractive (pulling q downward), which adds to gravity rather than opposing it. The signs must match so the force is repulsive and directed upward.
⚠️ Equilibrium and stability analysis is a recurring exam topic. Be ready to determine whether an equilibrium is stable or unstable by reasoning about what happens when the charge is nudged in each direction.
⚠️ Limiting-case checks are worth practising until they are automatic. Examiners often ask "does your answer make sense?" as a graded part of the problem.
⚠️ Know how to read a force-versus-position plot: a negative slope through the zero crossing means a restoring force (stable); a positive slope means an anti-restoring force (unstable).
⚠️ For the quadrupole, understand qualitatively why a single quadrupole produces an elliptical beam and why a pair is needed for full focusing. No calculation required, just physical reasoning.
True or false: If a charge is in electrostatic equilibrium, it is necessarily in stable equilibrium. (False)
In the suspended-charge problem, the equilibrium is unstable in the ______ direction. (z, i.e. perpendicular to the plane containing the three charges)
A potential energy minimum corresponds to ______ equilibrium. (stable)
True or false: A single electrostatic quadrupole can focus a beam in both transverse directions simultaneously. (False)
As m → 0 in the equilibrium formula, q → ______. (0)
Q: In the suspended-charge problem, why must q and Q have the same sign?
A: The electric force must push q upward to oppose gravity. Repulsion between like-signed charges produces an outward (upward) force. Opposite signs would cause attraction (downward), adding to gravity instead of balancing it.
Q: Explain in one sentence why the equilibrium is unstable in z but stable in x and y.
A: In x and y, displacing q changes the geometry so that the net Coulomb force has a component pointing back toward the equilibrium point, but in z, the force on either side of equilibrium points in the same direction as the displacement, so there is nothing to bring q back.
Q: A circular electron beam passes through a single quadrupole. Describe the cross-section of the beam when it exits.
A: The beam is no longer circular; it is elliptical, compressed along the axis where the field pushes electrons inward and elongated along the perpendicular axis where the field pushes them outward.
Q: Why are quadrupoles used in pairs in a CRT?
A: A single quadrupole focuses in one plane but defocuses in the other. A second quadrupole, rotated 90°, reverses the focusing and defocusing axes. Together the pair produces net focusing in both transverse directions, correcting beam imperfections so electrons hit the intended pixel.
Q: What is the purpose of checking limiting cases after deriving a formula?
A: Limiting cases test whether the formula behaves correctly in situations where the answer is already known by physical reasoning. If the formula fails a limiting case, it contains an error. This is one of the most efficient debugging tools in physics problem-solving.
The stability analysis here foreshadows Earnshaw's theorem, which states that a collection of point charges cannot produce a stable equilibrium for a free charge using electrostatic forces alone. You will encounter this again when studying electric potential and Laplace's equation.
The potential energy surface U(x, y, z) shown in the wrapup connects to the concept of electric potential (V), which you will study next week. Potential energy and electric potential are related by U = qV, so the landscape of V around charges directly determines where a charge "wants" to go.
Quadrupole focusing is a special case of charged-particle optics, which parallels the optics of light (lenses focus light, quadrupoles focus particle beams). This analogy becomes more explicit in advanced electromagnetism courses.
Coulomb force balance, electrostatic equilibrium, stable equilibrium, unstable equilibrium, restoring force, limiting cases, saddle point, potential energy surface, Earnshaw's theorem, electrostatic quadrupole, alternating-gradient focusing, strong focusing, linear charge density, beam optics, CRT, particle accelerator, P212, PHYS 212, forces between point charges