Source: P212 Discussion Worksheet, UIUC
Tags: 2-D motion, uniform force, kinematics, projectile motion analogy, dimensional analysis, charged particle deflection, parallel plates, P211 review, P212
Difficulty: Introductory (P211 review material) Prerequisites: P211 kinematics, Newton's second law, constant-acceleration equations.
This problem is a mechanics warm-up, but the physics maps directly onto a central P212 scenario: a charged particle deflected by a uniform electric field between parallel plates. The maths is identical, with the electric force qE replacing the generic force F. If you can solve this cleanly, you already know how to handle cathode-ray tube and capacitor-deflection problems later in the course.
You should be comfortable with constant-acceleration kinematics in two dimensions and with separating motion into independent x and y components before starting.
A particle enters a region of uniform force at constant horizontal speed, gets deflected vertically (like a ball thrown sideways off a cliff), and exits at some y-position and angle. The key results come from splitting the motion into an unaccelerated x-direction and a uniformly accelerated y-direction, then eliminating time.
Dimensional analysis
Checking whether a proposed formula has the correct physical units. If the units do not work out, the formula is guaranteed wrong, regardless of the algebra. Think of it as: a quick sanity filter that catches mistakes before you do any real calculation.
Limiting behaviour
Testing a formula by pushing a variable to an extreme value (zero, infinity, very large, very small) and checking whether the result makes physical sense. In simple terms, this means asking "if I crank this knob all the way up or down, does my answer do what I'd expect?"
Uniform force
A force that is constant in both magnitude and direction throughout the region of interest. The acceleration it produces is also constant. Think of it as: the gravitational force near Earth's surface, but possibly in a different direction.
Plate spacing (w)
The distance between the two parallel plates, measured perpendicular to the particle's initial direction of travel. The particle enters on the midplane y = 0, so it has w/2 of clearance above and below.
Plate length (L)
The extent of the plates in the x-direction, i.e. the horizontal distance over which the force acts.
The particle travels in the +x direction at constant speed v₀.
At x = 0 it enters the plate region. The plates run from x = 0 to x = L.
Between the plates a constant force F acts in the +y direction.
The particle enters at y = 0, midway between the plates (which sit at y = +w/2 and y = −w/2).
Outside the plates, no force acts.
Three candidate expressions are offered:
(1) y = F / [m(v₀ + L)] — wrong on units. v₀ + L adds a velocity to a length, which is meaningless.
(2) y = FL / (mv₀) — wrong on units. The numerator FL has units of N·m = J (energy), and the denominator mv₀ has units of kg·m/s (momentum), giving units of m/s, not metres.
(3) y = Fw² / (mv₀²) — correct units (check: N·m² / (kg·m²/s²) = (kg·m/s²)·m²·s²/(kg·m²) = m), but the formula uses w instead of L, which fails a limiting-behaviour test: increasing the plate length L should increase the deflection, yet this expression has no L dependence at all.
Dimensional analysis alone tells you the correct form must look like FL² / (mv₀²), because that is the only combination of the given variables that yields metres and that increases with F and L while decreasing with m and v₀. The exact numerical prefactor (which turns out to be 1/2) cannot be found by dimensions alone.
Split the motion into independent components.
x-direction (no force):
x(t) = v₀ t
Time to cross the plates: t_L = L / v₀
y-direction (constant force F):
Acceleration: a_y = F / m
y(t) = ½ a_y t² = (F / 2m) t²
Substitute t_L:
y_exit = FL² / (2mv₀²)
This confirms the dimensional-analysis prediction with the prefactor 1/2.
The particle strikes a plate when y_exit = w/2 (it enters at y = 0 and the nearest plate in the +y direction is at w/2).
Set FL² / (2mv₀²) = w/2 and solve for F:
F_max = mv₀²w / L²
Limiting-behaviour checks:
Larger w (wider gap) → larger F_max. Correct: more room means you can push harder.
Larger L (longer plates) → smaller F_max. Correct: the particle spends more time in the field, so less force is needed to reach the plate.
Larger v₀ → larger F_max. Correct: a faster particle spends less time between the plates, so it takes more force to deflect it to the edge.
The vertical velocity at exit:
v_y = a_y · t_L = (F/m)(L/v₀)
The horizontal velocity is unchanged: v_x = v₀.
tan θ = v_y / v_x = FL / (mv₀²)
Under the maximum-force condition from part (c), substitute F_max:
tan θ = w / L
Units check: w/L is dimensionless, as tan θ must be.
Limiting-behaviour check: if the plates are very long relative to their spacing (L ≫ w), the exit angle is small, which makes sense geometrically.
Quantity | Expression |
|---|---|
Time in plates | t_L = L / v₀ |
Exit y-coordinate | y = FL² / (2mv₀²) |
Maximum force | F_max = mv₀²w / L² |
Exit angle | tan θ = FL / (mv₀²) |
tan θ at F_max | tan θ = w / L |
This is exactly how a cathode-ray tube (CRT) steers an electron beam: horizontal and vertical deflection plates apply a uniform electric field to bend the beam to the desired spot on the screen. The same geometry appears whenever a charged particle passes through a parallel-plate capacitor.
Students sometimes add v₀ and L (as in candidate answer 1). Speed and length have different dimensions and cannot be added.
Forgetting that the numerical prefactor (the 1/2) matters when finding the maximum force. Dimensional analysis gives you the form, not the exact coefficient.
Assuming the particle accelerates in the x-direction. There is no x-component of force, so v_x stays at v₀ throughout.
Confusing the exit position (y) with the exit angle (θ). The position depends on L², while tan θ depends on L to the first power.
⚠️ Dimensional analysis and limiting-behaviour checks are tested repeatedly throughout P212. Practise them until they are automatic.
⚠️ This decomposition into independent perpendicular motions reappears every time a charge enters a uniform field (capacitor problems, ion optics, mass spectrometers).
⚠️ The relationship tan θ = w/L at maximum force is a clean, memorable result that examiners like to ask about.
True or false: the horizontal speed of the particle changes while it is between the plates.
Fill in the blank: the exit y-coordinate is proportional to L to the power of ____.
True or false: doubling the plate spacing doubles the maximum allowable force.
Fill in the blank: dimensional analysis can determine a formula up to a dimensionless ____.
True or false: after exiting the plates, the particle continues to accelerate upward.
Answers: 1. False. 2. Two. 3. True. 4. Prefactor (or constant). 5. False (no force acts outside the plates).
Q: A proton enters a parallel-plate region of length L at speed v₀. A uniform electric force F acts on it in the +y direction. Write the expression for the y-coordinate at exit, stating any assumptions.
A: y = FL² / (2mv₀²), assuming the plates are wide enough that the proton does not strike either plate and that fringe fields at the edges are negligible.
Q: How would you verify, without solving the full problem, that a proposed answer y = F²L / (m²v₀³) is incorrect?
A: Check units. F² has units (kg·m/s²)², L is m, m² is kg², v₀³ is (m/s)³. Working through: (kg²·m²/s⁴)(m) / (kg²·m³/s³) = 1/s, which is not a length. The expression fails dimensional analysis.
Q: Under what condition does the particle just barely clear the plates?
A: When y_exit = w/2, i.e. when F = mv₀²w / L².
Q: At maximum force, what is the exit angle in terms of w and L only?
A: tan θ = w / L.
This connects directly to P212's treatment of charged-particle motion in uniform electric fields (capacitor deflection problems). The same decomposition of motion into independent axes reappears in magnetic-force problems, though there the force depends on velocity, making the trajectory curved rather than parabolic. The dimensional-analysis technique introduced here is a tool you will use throughout the course.
Related Terms / Search Tags: parallel plate deflection, charged particle in uniform field, projectile motion analogy, constant force kinematics, dimensional analysis physics, limiting behaviour check, CRT deflection, capacitor particle trajectory, P212 UIUC, P211 review