The questions are on the Ampère's law test. Take it under real conditions before reading the mark scheme.
Quick answer: The Biot–Savart law gives the field of a current element, dB = (μ₀/4π) I dl × r̂/r², and integrating it yields B = μ₀I/2πd for a long wire and μ₀I/2R at a loop's centre. This mark scheme gives the model answers and the marks for the 8 questions of the AP Physics C test on Ampère's law, from Ampère’s law to toroid, with the equivalent answers to accept.
Award marks as shown. Accept answers within rounding and carry a correct method through from an earlier slip. Remove a mark once for a missing unit, and award a mark for each correct justification.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 12
Question 1 Ampère’s law (2 marks)
The line integral of B around a closed loop equals μ₀ times the net current through the loop, ∮B·dl = μ₀Ienc. It is useful when the symmetry of the current lets B be constant and tangent (or perpendicular) on the chosen loop.
Marking: 1 mark for the law; 1 mark for the symmetry condition.
Question 2 Long wire (2 marks)
B = μ₀I/(2πd) = (2 × 10⁻⁷)(6.0)/0.040 = 3.0 × 10⁻⁵ T.
Marking: 1 mark for the formula; 1 mark for the answer with unit.
Question 3 Coil (3 marks)
Centre: B = μ₀NI/(2R) = (4π × 10⁻⁷)(100)/(0.20) = 6.3 × 10⁻⁴ T. Axis: B = μ₀NIR²/[2(x² + R²)3/2] = 6.28 × 10⁻⁴ × (1/2)3/2 = 2.2 × 10⁻⁴ T.
Marking: 1 mark for the centre formula; 1 mark for the centre value; 1 mark for the axis value.
Question 4 Solenoid (3 marks)
n = 1500/0.50 = 3000 m⁻¹. B = μ₀nI = (4π × 10⁻⁷)(3000)(3.0) = 1.1 × 10⁻² T. Outside, B is approximately zero.
Marking: 1 mark for n; 1 mark for B; 1 mark for the outside field.
Question 5 Inside a thick wire (3 marks)
Inside: B = μ₀Ir/(2πR²) = (2 × 10⁻⁷)(6.0)(0.50 × 10⁻³)/(1.5 × 10⁻³)² = 2.7 × 10⁻⁴ T. Outside: B = μ₀I/(2πr) = (2 × 10⁻⁷)(6.0)/(3.0 × 10⁻³) = 4.0 × 10⁻⁴ T.
Marking: 1 mark for the enclosed-current idea; 1 mark for each value.
Question 6 Parallel wires (3 marks)
F/L = μ₀I₁I₂/(2πd) = (2 × 10⁻⁷)(12)(12)/0.050 = 5.8 × 10⁻⁴ N/m. The currents are in the same direction, so the force is attractive.
Marking: 1 mark for the formula; 1 mark for the value; 1 mark for attractive.
Question 7 Zero field on the line (2 marks)
The law has dB ∝ dl × r̂. On the line of the wire, dl is parallel to r̂, so the cross product is zero for every element, and the total field is zero.
Marking: 1 mark for the cross product; 1 mark for the parallel vectors.
Question 8 Toroid (2 marks)
B = μ₀NI/(2πr) = (2 × 10⁻⁷)(250)(0.80)/0.050 = 8.0 × 10⁻⁴ T.
Marking: 1 mark for the formula; 1 mark for the answer.
