Physics • Year 12 • Module 6: Electromagnetism • Lesson 5
Circular Motion in Magnetic Fields
Build HSC Band 5–6 extended-response technique on deriving and applying the circular-motion equations, comparing particle orbits, and evaluating the cyclotron principle.
1. Data + scenario: measuring the mass of an unknown ion (Band 5–6)
9 marks Band 5–6
Scenario. A scientist uses a mass spectrometer to identify an unknown singly charged positive ion (charge q = 1.60 × 10−19 C). She fires the ion into a uniform magnetic field of B = 0.40 T at a speed of v = 6.0 × 105 m/s, perpendicular to the field. The ion follows a circular arc and strikes a detector plate. The straight-line distance between the entry point and the detector strike equals the diameter of the circular orbit, measured as d = 0.374 m.
Q1. Analyse and evaluate the data above to identify the unknown ion. In your response you must:
- Derive the formula r = mv/qB starting from equating magnetic force to centripetal force.
- Use the measured diameter to calculate the orbital radius and hence calculate the mass of the ion.
- Show that the mass is consistent with one of the following candidates: carbon-12 ion (m = 1.993 × 10−26 kg), nitrogen-14 ion (m = 2.326 × 10−26 kg), or oxygen-16 ion (m = 2.656 × 10−26 kg).
- Calculate the period of the ion’s orbit and explain whether it would change if the ion’s speed were doubled.
- State one source of systematic error in this technique and one way to reduce it.
2. Experimental design — testing the independence of period from speed (Band 5–6)
8 marks Band 5–6
Research question. A student claims: “The period of a charged particle orbiting in a magnetic field must depend on its speed, because faster particles cover more distance.” Design an investigation using a magnetic field and an electron beam to test whether the orbital period is truly independent of speed.
Constraints: You have access to an electron gun (variable accelerating voltage 0–5 kV), a uniform magnetic field source (Helmholtz coils, adjustable), a timing device accurate to 1 × 10−10 s, and a phosphorescent screen to observe the beam. The investigation must be completed within a 2-hour laboratory session.
Q2. Design the investigation and present it in the format below.
- State a testable hypothesis including the independent and dependent variables.
- Identify controlled variables (at least two) and explain why each must be held constant.
- Describe the procedure in at least four numbered steps, including how the period will be measured and how speed will be varied.
- Explain what result would support and what result would falsify the hypothesis.
- State two limitations of the design and one improvement that could make the result more reliable.
Q1 — Sample Band 6 response (9 marks), annotated
Derivation of r = mv/qB (2 marks): The magnetic force on the ion is F = qvB (with velocity perpendicular to B). This force acts toward the centre of the circular path and provides the centripetal force: Fc = mv²/r. Setting these equal: qvB = mv²/r. Divide both sides by v: qB = mv/r. Rearrange: r = mv/qB. [1 mark for equating qvB to mv²/r; 1 mark for correct rearrangement to r = mv/qB]
Calculate orbital radius and mass (2 marks): The diameter equals twice the radius: r = d/2 = 0.374/2 = 0.187 m. From r = mv/qB, rearranging: m = qBr/v = (1.60 × 10−19 × 0.40 × 0.187) / (6.0 × 105). Numerator: 1.60 × 10−19 × 0.40 × 0.187 = 1.197 × 10−20. Divide by 6.0 × 105: m = 1.99 × 10−26 kg. [1 mark for r calculation; 1 mark for correct mass]
Ion identification (1 mark): The calculated mass (1.99 × 10−26 kg) matches the carbon-12 ion (m = 1.993 × 10−26 kg) to within rounding. The nitrogen-14 and oxygen-16 candidates have masses 17% and 33% larger respectively, so carbon-12 is the unambiguous match. [1 mark for correct identification with comparison reasoning]
Period calculation and speed independence (2 marks): T = 2πm/qB = 2π × 1.993 × 10−26 / (1.60 × 10−19 × 0.40) = 1.253 × 10−25 / 6.40 × 10−20 ≈ 1.96 × 10−6 s [1 mark]. If the speed were doubled, the period would remain unchanged because T = 2πm/qB contains no velocity term. A faster ion would travel in a larger circle but complete it in exactly the same time [1 mark].
Systematic error (2 marks): One valid systematic error: the ion may not enter the field exactly perpendicular to B, causing the path to be helical rather than circular and giving an incorrect radius measurement [1 mark]. Improvement: use a collimating slit or velocity selector to ensure the beam enters the field perpendicular to B, and place the detector slit in the same plane as the entry point to force measurement of the semicircular diameter [1 mark].
Marking summary (9 marks): 2 for derivation; 2 for correct calculation of r and m; 1 for correct identification of carbon-12 with comparison reasoning; 2 for period calculation and explanation of speed independence; 2 for systematic error and improvement.
Q2 — Sample Band 6 response (8 marks), annotated
Hypothesis (1 mark): If the orbital period T is independent of the electron’s speed, then increasing the accelerating voltage V (which increases speed) will not change the measured period T. Independent variable: accelerating voltage V (0–5 kV). Dependent variable: orbital period T.
Controlled variables (2 marks): (1) Magnetic field strength B (held constant by fixing the current through the Helmholtz coils); if B changed, T = 2πm/qB would change and confound the result [1 mark]. (2) The identity of the charged particles (electrons, fixed mass and charge); switching to a different particle type would change m/q and therefore T [1 mark].
Procedure (2 marks): (1) Set B to a fixed value using Helmholtz coils (e.g. calibrated to 5.0 × 10−3 T) and verify with a Hall probe. (2) Set the accelerating voltage to V1 = 1.0 kV. The electrons gain kinetic energy qV1 = ½mev2, so v = √(2qV/me); calculate v and use the observed circular orbit radius on the screen to verify. (3) Record the orbital radius r from the phosphorescent screen. Calculate the period T = 2πr/v. (4) Repeat for V = 2.0, 3.0, 4.0, 5.0 kV. Plot T vs V. If T is constant for all values of V, the hypothesis is supported. [2 marks for four clear, logically ordered steps including measurement method]
Support and falsification (1 mark): If T remains constant (or statistically indistinguishable) across all values of V, the data support the hypothesis. If T systematically increases or decreases with V, the hypothesis would be falsified [1 mark].
Limitations (2 marks): (1) At higher accelerating voltages the electron speed approaches a significant fraction of the speed of light; relativistic effects increase the effective mass and would cause T to increase slightly, introducing a systematic error not accounted for in the classical model [1 mark]. (2) The period is calculated indirectly from r and v rather than measured directly; measurement uncertainty in r from reading the phosphorescent screen limits the precision of T [1 mark].
Marking summary (8 marks): 1 = testable hypothesis with IV and DV; 2 = two controlled variables with justification; 2 = four-step procedure including measurement method; 1 = correct support/falsification statement; 2 = two valid limitations.