AQA A-Level Physics Paper 2, June 2023: Question 17

1 mark · Medium difficulty · Multiple Choice

Identify what the area under the curve of an electric field strength versus distance graph from the surface of a charged sphere to infinity represents.

Practise this question

Question

A multiple-choice question showing a graph of electric field strength E on the vertical axis plotted against distance from the centre of the sphere r on the horizontal axis. A dashed line marks the radius of the sphere, R. For r greater than or equal to R, a curve decreases smoothly towards zero as r increases, with the area under the curve between R and infinity shaded grey. Candidates are asked what this total shaded area represents, with choices: A the capacitance of the sphere, B the charge held on the sphere, C the electric potential of the sphere, D the energy needed to remove an electron from the sphere.
Question text

17 The graph shows the variation of electric field strength E surrounding a charged

sphere of radius R. The distance from the centre of the sphere is r.

The total area under the curve from R to infinity is

[1 mark]

A the capacitance of the sphere.

B the charge held on the sphere.

C the electric potential of the sphere.

D the energy needed to remove an electron from the sphere.

Mark scheme

Show the mark scheme Mark scheme showing question 17 with correct answer C, stating 'the electric potential of the sphere.'

17 C the electric potential of the sphere.

How to answer it

Area Under an Electric Field Strength–Distance Graph

📌 What this question tests

This question assesses your understanding of graphical relationships in non-uniform radial electric fields, specifically:

  • The relationship between electric field strength ( E ) and electric potential ( V ).
  • Physical interpretation of the area under an E – r curve ( E = -ΔV / Δr ).
  • The definition of electric potential at the surface of a charged sphere relative to infinity where V = 0 .
Question 17 • Multiple Choice [1 Mark]

Identifying the Physical Quantity Represented by the Area

Relating Field Strength E to Potential V

✅ Correct Answer

C: the electric potential of the sphere.

Award [1 mark] for selecting option C.

🧠 Exam Technique: Graphical Calculus

Always inspect the gradient and area using calculus definitions:

  • Gradient of a y vs x graph = dy / dx
  • Area under a y vs x graph = ∫ y dx

Here, the area is ∫ E dr . Check the units: (N C⁻¹) × m = N m C⁻¹ = J C⁻¹ = V (volts).

📐 Step-by-Step Mathematical Derivation

  1. Recall the field-potential relationship:
    The electric field strength is equal to the negative potential gradient:
    E = - dV / dr or ΔV = - ∫ E dr
  2. Integrate from the surface (r = R) to infinity (r = ∞):
    Area = ∫R∞ E dr
  3. Substitute the field of a point charge / conducting sphere:
    E = Q / (4πε₀r²)
    Area = ∫R∞ [Q / (4πε₀r²)] dr = [-Q / (4πε₀r)]R∞ = 0 - (-Q / (4πε₀R)) = Q / (4πε₀R)
  4. Compare with standard definitions:
    The electric potential V on the surface of a sphere of radius R is precisely:
    V = Q / (4πε₀R)
    Since V = 0 at infinity, the area represents the magnitude of the electric potential of the sphere.

💡 Key Knowledge: Radial Fields

  • Inside a charged conducting sphere: E = 0 , while V is constant and equal to its surface value.
  • At surface & beyond (r ≥ R): The sphere acts as if all its charge Q is concentrated at its centre.
  • Potential Definition: Work done per unit positive charge in bringing a small test charge from infinity to that point. Hence:
    V = - ∫∞r E dr = ∫r∞ E dr

❌ Common Distractors & Why They Are Wrong

  • A (Capacitance): Capacitance is C = Q / V . For an isolated sphere, C = 4πε₀R , which is a constant geometry property, not the area under an E–r graph.
  • B (Charge held on sphere): Charge Q has units of coulombs (C). The area has units of (V m⁻¹) × m = V (volts or J C⁻¹), so dimensionally it cannot be charge.
  • D (Energy needed to remove an electron): Energy requires multiplying electric potential by charge ( ΔEₚ = qΔV ). The area is work done per unit charge, not total energy for an electron.

Topics

Physics · 3.7 Fields and their consequences (A-level only)

Question and mark scheme from the AQA A-Level Physics examination, Paper 2, June 2023. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.