AP · AP C:E&M · free response
AP Physics C: Electricity and Magnetism · Question 7
AP Physics C: Electricity and Magnetism · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- discriminating
- Marks
- 14
- Topics
- 2
- Answer
- Complete
An insulating sphere of radius R carries a volume charge density that varies with distance from the centre as ρ(r) = ρ₀ r/R for r ≤ R, where ρ₀ is a positive constant. There is no charge outside the sphere.
- (a)
Derive Derive an expression for the total charge Q on the sphere.
3 marks - (b)
Derive Using Gauss's law, derive an expression for the magnitude of the electric field at a distance r from the centre, for r < R.
4 marks - (c)
Derive Derive an expression for the magnitude of the electric field at a distance r from the centre, for r > R, and verify that your two expressions agree at r = R.
3 marks - (d)
Derive Derive an expression for the electric potential at the centre of the sphere, taking the potential to be zero at infinity.
4 marks
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(a)
Derive Derive an expression for the total charge Q on the sphere.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: uses a spherical shell of volume dV = 4πr²dr as the element of integration
- 2
1 point: Q = ∫₀^R ρ₀(r/R)4πr² dr = (4πρ₀/R)∫₀^R r³ dr
- 3
1 point: Q = πρ₀R³
(b)
Derive Using Gauss's law, derive an expression for the magnitude of the electric field at a distance r from the centre, for r < R.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: chooses a concentric spherical Gaussian surface of radius r and states that E is radial and constant on it, so ∮E·dA = E(4πr²)
- 2
1 point: computes the enclosed charge by integrating, q_enc = (4πρ₀/R)(r⁴/4) = πρ₀r⁴/R
- 3
1 point: sets E(4πr²) = q_enc/ε₀
- 4
1 point: E = ρ₀r²/(4ε₀R)
(c)
Derive Derive an expression for the magnitude of the electric field at a distance r from the centre, for r > R, and verify that your two expressions agree at r = R.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: outside the sphere the whole charge is enclosed, so E = Q/(4πε₀r²) = ρ₀R³/(4ε₀r²)
- 2
1 point: substitutes r = R into the inside expression to get ρ₀R/(4ε₀)
- 3
1 point: substitutes r = R into the outside expression to get the same value, confirming the field is continuous at the surface
(d)
Derive Derive an expression for the electric potential at the centre of the sphere, taking the potential to be zero at infinity.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
1 point: V(0) = ∫₀^∞ E dr, splitting the integral at r = R
- 2
1 point: outside contribution ∫_R^∞ ρ₀R³/(4ε₀r²) dr = ρ₀R²/(4ε₀)
- 3
1 point: inside contribution ∫₀^R ρ₀r²/(4ε₀R) dr = ρ₀R²/(12ε₀)
- 4
1 point: V(0) = ρ₀R²/(4ε₀) + ρ₀R²/(12ε₀) = ρ₀R²/(3ε₀)
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