AQA A-Level Physics Paper 2, June 2023: Question 10

AQA A-Level Physics • Gravitational & Electric Fields

Comparing Electric and Gravitational Forces Between Protons

What this question tests

This question assesses your ability to apply Coulomb's Law and Newton's Law of Gravitation to fundamental particles, recognize inverse-square law cancellations, retrieve physical constants from the data booklet, and perform order-of-magnitude estimations under timed multiple-choice conditions.

Question 10 (1 Mark)

Ratio of Electrostatic Force to Gravitational Force

Two protons separated by a distance of 1 × 10⁻⁹ m

✅ Correct Answer

Option C: 10³⁶

Mark Scheme: Award 1 mark for selecting C (10³⁶).

The electrostatic repulsion between two protons is approximately 36 orders of magnitude stronger than their gravitational attraction.

💡 Key Knowledge

Coulomb's Law: F_E = (1 / 4πε₀) × (q₁q₂ / r²)
Newton's Law of Gravitation: F_G = G × (m₁m₂ / r²)
Distance independence: Because both obey an inverse-square law ( ∝ 1/r² ), the separation distance r completely cancels out in the ratio.
Constants from Data Booklet:
- 1 / (4πε₀) ≈ 8.99 × 10⁹ N m² C⁻²
- G ≈ 6.67 × 10⁻¹¹ N m² kg⁻²
- Proton charge e = 1.60 × 10⁻¹⁹ C
- Proton rest mass m_p = 1.67 × 10⁻²⁷ kg

📐 Step-by-Step Calculation

Set up the ratio:
Ratio = F_E / F_G = [ (1 / 4πε₀) × (e² / r²) ] / [ G × (m_p² / r²) ]
Cancel separation r² :
Ratio = [ (1 / 4πε₀) × e² ] / [ G × m_p² ]
Notice: The given distance of 1 × 10⁻⁹ m is not needed!
Estimate powers of ten (Mental / Quick Check):
- Numerator power: 10⁹ × (10⁻¹⁹)² = 10⁹ × 10⁻³⁸ = 10⁻²⁹
- Denominator power: 10⁻¹¹ × (10⁻²⁷)² = 10⁻¹¹ × 10⁻⁵⁴ = 10⁻⁶⁵
- Power ratio: 10⁻²⁹ / 10⁻⁶⁵ = 10³⁶
Full calculation confirmation:
Numerator = (8.99 × 10⁹) × (1.60 × 10⁻¹⁹)² ≈ 2.30 × 10⁻²⁸
Denominator = (6.67 × 10⁻¹¹) × (1.67 × 10⁻²⁷)² ≈ 1.86 × 10⁻⁶⁴
Ratio = (2.30 × 10⁻²⁸) / (1.86 × 10⁻⁶⁴) = 1.24 × 10³⁶ ≈ 10³⁶

🧠 Exam Technique & Time Savers

Spot redundant data: Examiners frequently include values (like the separation 1 × 10⁻⁹ m ) to see if you instinctively spot the cancellation. Don't waste precious minutes squaring and dividing by 1 × 10⁻⁹ twice!
Order-of-magnitude estimation: When options differ vastly by powers of 10 ( 10¹⁸, 10²⁸, 10³⁶, 10⁴⁵ ), you only need to track the exponents: 9 - 38 - (-11 - 54) = -29 - (-65) = +36 . This solves the question in under 20 seconds.

❌ Common Errors & Pitfalls

Forgetting to square the charges or masses: A common slips is calculating e / m_p instead of e² / m_p² .
Sign errors in exponents: Subtracting negative powers incorrectly in the denominator: -29 - (-65) = +36 , not -94 .
Using electron mass instead of proton mass: If you use m_e = 9.11 × 10⁻³¹ kg , you will get a ratio near 10⁴²–10⁴³ , which might tempt you to select D (10⁴⁵). Ensure you pick the proton mass!