AQA A-Level Physics Paper 2, June 2023: Question 14

📋 Specification Focus

This multiple-choice question assesses your ability to determine resultant electric field strengths from multiple point charges by applying:

Coulomb's Law definition of Electric Field Strength: E = Q / (4πε₀r²) .
Vector addition of fields: Recognising that electric field is a vector quantity with direction defined as the force per unit positive test charge (away from positive charges, towards negative charges).
Symmetry arguments: Pairing diametrically opposite charges to quickly cancel components and avoid lengthy trigonometry.
Radius vs. diameter substitutions: Correctly substituting r = d / 2 into inverse-square expressions.

Question 14 • 1 Mark

Option C: 2Q / (πε₀d²)

Mark Scheme: 1 mark awarded for selecting C.

Field of a Point Charge:
E = Q / (4πε₀r²)
Field Direction:
- From +Q: directed radially outward (away from charge).
- From −Q: directed radially inward (towards charge).
Distance from Centre: All 6 spheres sit on a circle of diameter d , so each sphere is at distance r = d / 2 from the centre.

Identify Diametrically Opposite Pairs:
The 6 charges are spaced at 60° intervals around the circle. Group them as 3 opposing pairs across lines passing through the centre:
- Pair 1 (Top-Left & Bottom-Right): −Q and −Q . Both fields point towards their respective charges with equal magnitude. Being opposite in direction, they cancel out completely ( E_net = 0 ).
- Pair 2 (Top-Right & Bottom-Left): +Q and +Q . Both fields point away from their respective charges with equal magnitude. Being opposite in direction, they cancel out completely ( E_net = 0 ).
- Pair 3 (Left & Right): −Q (left) and +Q (right).
  • Field from +Q points away from the right charge (to the left).
  • Field from −Q points towards the left charge (to the left).
  Both vectors point in the same direction (towards the left), so their magnitudes add together!
Calculate the Field Magnitude of One Sphere:
Substitute r = d / 2 into the field equation:
E₁ = Q / [4πε₀(d / 2)²] = Q / [4πε₀(d² / 4)] = Q / (πε₀d²)
Combine the Uncancelled Fields:
Total resultant field:
E_total = E(+Q) + E(−Q) = E₁ + E₁ = 2 × [Q / (πε₀d²)] = 2Q / (πε₀d²)

Look for symmetry first: In 1-mark multiple choice questions, never resolve all 6 vectors into components using trigonometry. Always check opposite pairs for cancellation.
Sign check: Remember that two like charges opposite each other cancel, but two opposite charges opposite each other reinforce!
Radius vs. Diameter shortcut: Because (d/2)² = d²/4 , the factor of 4 cancels immediately with the 4 in 4πε₀ , giving a single charge field of Q / (πε₀d²) .

Selecting Option A (0): Students see a symmetric hexagonal arrangement and assume everything cancels out by symmetry without checking the signs of opposite charges.
Selecting Option B: Forgetting that both the +Q and −Q sphere contribute to the field towards the left, forgetting to multiply by 2.
Selecting Option D: Forgetting that (1/2)² = 1/4 cancels out the 4 in 4πε₀ , leading to an extra unwanted factor of 2.
Confusing Potential with Field Strength: Electric potential is a scalar (where total potential would be ΣV = 0 because there are 3 positive and 3 negative charges). Electric field is a vector, so directions matter!

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