University Physics II · topic UPII-08.03
Charged-particle motion in uniform fields
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Charged-particle orbits in uniform fields: predict a finite change
In a uniform magnetic field a charged particle travels at constant speed along a helix whose axis lies along B, circling at the cyclotron frequency while coasting steadily in the field direction. Use the supplied topic model Y_out = 5.686*vperp/b, where Y_out is ORBIT RADIUS r (mm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Field strength B, measured in mT. Keep vperp (Perpendicular speed v⊥) = 2 ×10⁶ m s⁻¹; vpar (Parallel speed v∥) = 1.2 ×10⁶ m s⁻¹ fixed. For this exercise use only 2.31875 mT ≤ b ≤ 4.68125 mT on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at b = 2.8385 mT and 3.16925 mT. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δb across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
ORBIT RADIUS r against b
Predictions from the supplied Charged-particle orbits in uniform fields model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ORBIT RADIUS r / mm
Horizontal axis (x): b: Field strength B / mT
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Field strength B / mT | ORBIT RADIUS r / mm |
|---|---|
| 2.31875 | 4.904366576819407 |
| 2.9093750000000003 | 3.908743286788399 |
| 3.5 | 3.249142857142857 |
| 4.090625 | 2.780015278838808 |
| 4.68125 | 2.4292656875834444 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of b separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At b = 2.8385 mT, substitute into Y_out = 5.686*vperp/b: Y₀ = 4.00634 mm.
- At b = 3.16925 mT, the same fixed-parameter model gives Y₁ = 3.58823 mm.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.418111 mm; the input change is Δb = 0.33075 mT.
- Average sensitivity is ΔY_out/Δb = -1.26413 mm per mT. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 4.00634 mm, Y₁ = 3.58823 mm; ΔY_out = -0.418111 mm; average sensitivity = -1.26413 mm per mT.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Charged-particle orbits in uniform fields: infer an input with uncertainty
In a uniform magnetic field a charged particle travels at constant speed along a helix whose axis lies along B, circling at the cyclotron frequency while coasting steadily in the field direction. Use the supplied topic model Y_out = 5.686*vperp/b, where Y_out is ORBIT RADIUS r (mm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Field strength B, measured in mT. Keep vperp (Perpendicular speed v⊥) = 2 ×10⁶ m s⁻¹; vpar (Parallel speed v∥) = 1.2 ×10⁶ m s⁻¹ fixed. For this exercise use only 2.31875 mT ≤ b ≤ 4.68125 mT on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 3.0245686 mm, with a bounded reading uncertainty ±0.089295 mm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00023625 mT, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
ORBIT RADIUS r against b
Predictions from the supplied Charged-particle orbits in uniform fields model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ORBIT RADIUS r / mm
Horizontal axis (x): b: Field strength B / mT
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Field strength B / mT | ORBIT RADIUS r / mm |
|---|---|
| 2.31875 | 4.904366576819407 |
| 2.9093750000000003 | 3.908743286788399 |
| 3.5 | 3.249142857142857 |
| 4.090625 | 2.780015278838808 |
| 4.68125 | 2.4292656875834444 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(b) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(2.31875) = 4.90437 mm and Y_out(4.68125) = 2.42927 mm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives b_line = 2.31875 + (3.02457 − 4.90437)(4.68125 − 2.31875)/(2.42927 − 4.90437) = 4.11303 mT. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives b = 3.75988 mT; substituting back gives Y_out = 3.0245686 mm.
- The interpolation error in the input is |b_line − b| = 0.353154 mT. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.9352736 mm and 3.1138636 mm. Sort the roots even if the response decreases: 3.65205 mT ≤ b ≤ 3.87426 mT.
- The lower and upper input deviations are 0.10782 mT and 0.11438 mT. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer b ≈ 3.75988 mT; compatible interval [3.65205, 3.87426] mT. Endpoint interpolation gives 4.11303 mT.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.