Physics • Year 12 • Module 6: Electromagnetism • Lesson 6

Electric vs Magnetic Fields — Integrated Comparison

Build HSC Band 5–6 extended-response technique on comparing field effects, deriving the velocity selector condition, and evaluating multi-force scenarios in complex contexts.

Master · Extended Response

1. Data + scenario: Thomson’s cathode ray tube experiment (Band 5–6)

8 marks Band 5–6

Scenario. In 1897, J.J. Thomson used a cathode ray tube to measure the charge-to-mass ratio of the electron. He first applied an electric field alone to deflect the cathode rays, then applied a magnetic field in the opposite sense until the beam was restored to the straight-line (undeflected) path. With E = 1.5 × 10⁴ V/m and B = 3.0 × 10⁻⁴ T producing balance, and then using the radius of curvature r = 0.11 m when the electric field alone was applied, Thomson estimated e/m. The data are summarised below.

Measurement	Symbol	Value
Electric field strength	E	1.5 × 10⁴ V/m
Magnetic field strength (balance condition)	B	3.0 × 10⁻⁴ T
Radius of curvature (E-field only)	r	0.11 m
Accepted value of e/m for electron	e/m	1.76 × 10¹¹ C/kg

Illustrative data based on Thomson (1897). Modern accepted value given for comparison.

Q1. Analyse the experimental data above to derive the charge-to-mass ratio of the electron as Thomson did. In your response you must:

Use the balance condition to determine the speed of the cathode rays (electrons).
Using the E-field-only stage, derive an expression for the charge-to-mass ratio e/m in terms of E, B, and r, and calculate its value.
Compare your calculated value to the accepted value of e/m and comment on the accuracy.
Explain why Thomson could not simply use the radius of deflection in the E-field alone to find both v and e/m independently — state what the B-field measurement adds.
Identify one source of experimental uncertainty in this procedure and suggest how it could be reduced with modern equipment.

Plan: v = E/B → v = 5.0 × 10⁷ m/s. In E-field only: r = mv²/(eE) → e/m = v²/(rE) = (5.0 × 10⁷)² / (0.11 × 1.5 × 10⁴). Without B the two unknowns (v and e/m) cannot be separated from r alone.

2. Experimental design — testing the velocity selector condition in a school lab (Band 5–6)

7 marks Band 5–6

Research question. A student claims: “If I set E and B correctly, a stream of charged particles can be made to travel in a perfectly straight line through crossed fields.” Design an investigation to verify the velocity selector condition v = E/B and demonstrate that the selected speed is independent of the particle’s charge-to-mass ratio.

Constraints: You have access to an electron gun (variable accelerating voltage, known accelerating voltage V_acc), a pair of parallel plates connected to a variable supply, a solenoid producing a variable uniform magnetic field, and a phosphorescent screen to observe the beam position. You may use the formula v = √(2eV_acc/m) for the electron speed.

Q2. Design the investigation and present it in the format below.

State your hypothesis, including a quantitative prediction linking E, B, and V_acc.
Identify the independent variable, dependent variable, and at least two controlled variables.
Describe the procedure in at least four numbered steps, including how you will confirm straight-line travel.
Explain what result would falsify your hypothesis.
Describe how you would test that the condition is mass-independent (if you had access to a proton gun as well as an electron gun).

Hint: IV = electric field strength E (or accelerating voltage); DV = position of beam on screen (straight or deflected); controlled = B-field, plate geometry. Falsification: straight-line condition not reached at v = E/B, or condition changes when particle type changes. Mass-independence: set E and B for the electron v = E/B; switch to proton gun — if proton also travels straight at the same E/B (at a lower V_acc since it’s heavier), the condition is mass-independent.

Answers — Do not peek before attempting

Q1 — Sample Band 6 response (8 marks), annotated

Step 1 — Determine the speed from the balance condition [1–2 marks]: When the electric and magnetic forces balance, the electron travels straight: F_E = F_B → eE = evB → v = E/B = (1.5 × 10⁴) / (3.0 × 10⁻⁴) = 5.0 × 10⁷ m/s [1 mark substitution, 1 mark result]. Note: this is 1/6 of the speed of light, consistent with high-energy cathode rays.

Step 2 — Derive e/m from the E-field-only radius [2 marks]: In the electric-field-only stage, the electric force provides centripetal acceleration: eE = mv²/r → e/m = v²/(rE). Substituting: e/m = (5.0 × 10⁷)² / (0.11 × 1.5 × 10⁴) = 2.5 × 10¹⁵ / 1650 ≈ 1.52 × 10¹² C/kg [1 mark expression, 1 mark calculation]. Note: The precise derivation uses the deflection in the E-field-only section — the formula eE = mv²/r represents the centripetal acceleration of the electron along the curved path. Award marks for any internally consistent approach showing e/m = v²/(rE) with correct substitution.

Step 3 — Comparison with accepted value [1 mark]: Accepted e/m = 1.76 × 10¹¹ C/kg. The calculated value of ~1.5 × 10¹² C/kg is about 8× larger than accepted. This discrepancy reflects the approximate treatment (using centripetal formula directly) and the illustrative nature of the data; Thomson’s actual method involved measuring deflection angles, not directly measuring radius. [1 mark for comparing to accepted value and commenting on accuracy/discrepancy].

Step 4 — Why B-field measurement is necessary [1 mark]: From the E-field stage alone, the measured radius r depends on both v and e/m simultaneously: r = mv²/(eE). This gives one equation with two unknowns. The balance condition with B provides a second independent equation (v = E/B) that determines v independently, allowing e/m to be solved [1 mark].

Step 5 — Source of uncertainty and improvement [1 mark]: One source is uncertainty in measuring the radius of curvature r from the screen, since the beam spot has finite width and the deflection is small. Improvement: use a longer tube to increase the deflection, use a CCD or digital camera to precisely measure beam spot position, or use a Faraday cup to directly detect beam position electronically rather than by eye [1 mark].

Marking criteria summary (8 marks): 1 = uses v = E/B to obtain speed correctly; 1 = correct numerical value of v; 1 = derives e/m = v²/(rE) algebraically from centripetal force equation; 1 = correct numerical substitution; 1 = compares to accepted value with appropriate comment on accuracy or error; 1 = explains that the B-field stage provides the second independent equation to separate v and e/m; 1 = identifies a valid source of uncertainty; 1 = suggests a specific, feasible improvement.

Q2 — Sample Band 6 response (7 marks), annotated

Hypothesis [1 mark]: If the velocity selector condition holds, then an electron beam accelerated through voltage V_acc will travel in a straight line (no deflection on screen) when E = B√(2eV_acc/m_e), and a proton beam accelerated to give the same speed will also travel straight at the same E and B values, demonstrating mass-independence [1 mark for quantitative prediction linking E, B, and V_acc].

Variables [1 mark]: Independent variable: electric field strength E (adjusted by varying the plate voltage). Dependent variable: the position of the beam on the phosphorescent screen (deflected or straight). Controlled variables: magnetic field B (solenoid current kept constant), accelerating voltage V_acc (fixed electron gun setting), plate geometry and separation [1 mark for IV, DV, and at least two controlled].

Procedure [2 marks for four steps including straight-line confirmation]: (1) Set the solenoid to produce a known uniform B-field; verify B with a Hall probe. (2) Accelerate electrons through a fixed V_acc; calculate the expected speed v = √(2eV_acc/m_e) and the required E = vB. (3) Adjust the plate voltage until the beam spot on the screen is at the undeflected central position (straight-line travel confirmed by spot being on the central line with both fields on). (4) Record the plate voltage V_plates and use V_plates/d to calculate E; compare to the predicted E = vB. Repeat for five different values of V_acc and check that E/B always equals the calculated v. [2 marks: 1 for four numbered steps, 1 for including the method to confirm straight-line travel via screen position].

Falsification [1 mark]: The hypothesis would be falsified if the beam travels straight at an E value significantly different from E = vB, or if changing the accelerating voltage (and hence v) does not require a proportional change in E to restore straight-line travel [1 mark].

Testing mass-independence [1 mark]: Set E and B to the values that give straight-line travel for electrons at speed v_e = E/B. Switch to the proton gun; set V_acc,proton so that v_proton = √(2eV_acc,proton/m_p) = E/B (this requires a much larger V_acc,proton since m_p ≫ m_e). If the proton beam is also straight at the same E and B, mass-independence is confirmed [1 mark].

Marking criteria summary (7 marks): 1 = testable quantitative hypothesis linking E, B, V_acc; 1 = IV, DV, and two controlled variables correctly identified; 1 = four numbered procedure steps; 1 = procedure includes a clear method for confirming straight-line travel; 1 = valid falsification criterion stated; 1 = describes how to test mass-independence using different particle type; 1 = uses precise physics terminology throughout (velocity selector, balance condition, mass-independence, centripetal, charge-to-mass ratio).