Lessons 3 · 1 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 11, 1st edition. Niso Poligraf Publishing House, Tashkent, 2018
3
The magnetic field of a straight current-carrying wire, a loop and a coil
Textbook: pp. 10–12
GoalCalculate the magnetic induction of a straight wire, a loop and a coil with the formulas, and apply the right-hand screw rule and the superposition principle.
New words
right-hand screw rule · parma qoidasisuperposition principle · superpozitsiya prinsipimagnetic constant · vakuumning magnit doimiysiloop and coil · halqa va g‘altak
Explanation
Around a straight current-carrying wire the field lines are circles centred on the wire and lying in planes perpendicular to it; their direction follows the right-hand screw rule: if the screw advances in the direction of the current, the turning of its handle gives the direction of the lines. At distance d from an infinitely long straight current B = μ₀I/(2π·d), where μ₀ = 4π·10⁻⁷ N/A² is the magnetic constant of vacuum. At the centre of a circular current of radius R, B = μ₀I/(2R), and at the centre of a coil of N turns B = μ₀NI/(2R). The resultant field of several currents is the vector sum of the fields of the separate currents: B = B₁ + B₂ + … (superposition principle). The Biot–Savart–Laplace law gives the induction ΔB produced by a small element Δl of a conductor of any shape; the formulas above are its results.
Worked examples
A long straight wire carries I = 10 A. At d = 5 cm from it B = μ₀I/(2πd) = 2·10⁻⁷·10/0.05 = 4·10⁻⁵ T = 40 μT (μ₀/2π = 2·10⁻⁷ N/A²).
Two long parallel wires are 10 cm apart and carry 10 A each in opposite directions. At the midpoint d = 5 cm from each: B₁ = B₂ = 40 μT, and they point the same way, so B = 80 μT. If the currents were in the same direction, the fields would cancel at the midpoint: B = 0.
Class activity
In the classroom, only with the teacher: watch a compass needle deflect near a wire briefly connected to a battery. The wire can get hot, so students never create short circuits themselves; experiments with household mains are strictly forbidden. Practise the right-hand screw rule with your hand.
Practice
1
How does the magnetic induction of a straight current change if the distance is doubled?
It halves (B ∝ 1/d).
2
A straight wire carries I = 20 A. What is B (in μT) at 10 cm from it? (μ₀/2π = 2·10⁻⁷ N/A²)
40
3
A coil with N = 50 turns and radius R = 10 cm carries I = 2 A. Find B at the centre (μ₀ = 4π·10⁻⁷ N/A²).
B = μ₀NI/(2R) = 2π·10⁻⁴ T ≈ 6.3·10⁻⁴ T
4
Why does a multi-turn coil give a stronger field at its centre than a single loop?
Every turn produces a field in the same direction and their inductions add: B = μ₀NI/(2R).