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Magnetic field questions on wires, loops and solenoids usually test one decision: Biot-Savart or Ampère? Pick the wrong one and you either cannot finish or get a wrong answer that looks right. This post shows how to choose, works through the standard fields with numbers, and lists the mistakes to avoid. For the force on a current in a field, see our separate post on the right-hand rule for magnetic force.
Which magnetic field results does the syllabus list?
The JEE (Main) 2026 syllabus lists the Biot-Savart law applied to a circular loop, and Ampère's law applied to an infinitely long straight wire and a solenoid. The JEE (Advanced) 2026 brochure lists both laws and the field near a straight wire, along the axis of a circular coil and inside a long solenoid. NCERT Class 12 Chapter 4 derives these in sections 4.4 to 4.7, and notes that Ampère's law is to Biot-Savart what Gauss's law is to Coulomb's law. Check your year's bulletin.
Why do students struggle with Ampère's law?
Wallace and Chasteen (2010) studied junior-level physics students through written questions and interviews. Some struggled to connect the current enclosed by an Ampèrian loop to the field, and some did not use what they knew about the field's direction and symmetry when solving problems. The authors read this as evidence that students did not see the integral in Ampère's law as a sum. Guisasola and colleagues (2008) found similar weak reasoning with Ampère's law and Gauss's law among university students. If it happens to physics majors, it is worth slowing down on.
When should you use each law?
Ampère's law works when you can draw a loop on which B is constant in size and parallel to the loop (or perpendicular, contributing nothing). That means a long straight wire, a long solenoid, a toroid, or a thick cylinder of current.
Biot-Savart works for any shape but needs an integral. Use it for a circular loop, an arc, or a finite straight wire.
Worked examples with numbers
Use μ0 = 4π × 10⁻⁷ T m/A, so μ0/(4π) = 10⁻⁷ and μ0/(2π) = 2 × 10⁻⁷.
Long straight wire (Ampère). I = 10 A, r = 5.0 cm. A circle round the wire gives B × 2πr = μ0I, so B = μ0I/(2πr) = 2 × 10⁻⁷ × 10/0.05 = 4.0 × 10⁻⁵ T. The direction follows the right-hand grip rule.
Centre of a circular coil (Biot-Savart). NCERT's exercise 4.1 has 100 turns of radius 8.0 cm carrying 0.40 A. Every element is perpendicular to the line to the centre and at the same distance, so B = μ0NI/(2R) = (4π × 10⁻⁷ × 100 × 0.40)/(0.16) ≈ 3.1 × 10⁻⁴ T.
On the axis of a loop. For one turn of radius R = 0.10 m with I = 5.0 A, B = μ0IR²/[2(R² + x²)^(3/2)]. At the centre (x = 0) B = 3.14 × 10⁻⁵ T. At x = 0.10 m, R² + x² = 0.02 m², and B ≈ 1.11 × 10⁻⁵ T, about 0.35 of the centre value, which matches the factor 1/(2√2).
Long solenoid (Ampère). A solenoid 0.25 m long has 500 turns, so n = 2000 turns per metre. With I = 2.0 A, B = μ0nI = 4π × 10⁻⁷ × 2000 × 2.0 ≈ 5.0 × 10⁻³ T inside, and nearly zero outside.
What mistakes show up most often?
Ampère's law on a finite wire or a loop. The law is still true, but you cannot pull B out of the integral, so it gives you nothing useful.
Total turns in place of turns per metre. In B = μ0nI, n is turns per unit length. 500 turns on 0.25 m is 2000 per metre, not 500.
Forgetting N for a coil. A coil of N turns gives N times the single-turn field.
Mixing 2π and 4π. Write μ0/(2π) = 2 × 10⁻⁷ and μ0/(4π) = 10⁻⁷ at the top of your page.
Direction last. Find the direction with the right-hand grip rule before the magnitude, especially when adding fields from two wires.
A practice routine
Take eight current shapes from past papers. For each, write only "Ampère" or "Biot-Savart" and one line explaining why.
Solve the eight fully. For two of them, also do the problem with the other law and see where it breaks.
Add two superposition problems: two parallel wires, and a loop with a straight wire through its centre.
The same symmetry idea drives Gauss's law problems; doing a few of each in one session helps you see the pattern.
If you are not sure which law fits, Fermi's AI tutor notices where you are stuck and decides whether to explain, ask or show, and it can draw the figure as it goes, so you see the symmetry rather than take it on trust.
Tonight, write the eight "Ampère or Biot-Savart" decisions before you solve any of them.
