PHYC10004 Chap.7 Magnetic Fields from Currents: Biot-Savart and Ampere
Magnetic Fields from Currents: Biot-Savart and Ampere
Currents create magnetic fields, and PHYC10004 teaches two ways to calculate them. The Biot-Savart law gives the field of each small element of current and always works, but often needs careful integration; it yields the field of a long straight wire, B = μ₀i/2πr, and of a semicircular arc at its centre, μ₀i/4R.
Ampere's law, which relates the line integral of B around a closed loop to the current through it, plays the role Gauss's law played for electric fields.
Where symmetry makes the field constant along a well-chosen loop, Ampere's law gives the field in a line or two: inside and outside a thick wire, between the conductors of a coaxial cable, and inside a long solenoid, B = μ₀ni.
The chapter also covers forces between parallel currents, which attract when the currents run the same way, Gauss's law for magnetism, and why a ferromagnetic core multiplies a solenoid's field. Past papers repeatedly ask for a derivation of the wire or solenoid field with the assumptions stated.
What this chapter covers
- 01
The Biot-Savart Law
- 02
Field of a Long Wire
- 03
Field of an Arc or Loop
- 04
Ampere's Law
- 05
Choosing an Amperian Loop
- 06
Solenoids and Toroids
- 07
Coaxial Conductors
- 08
Forces Between Parallel Wires
- 09
Magnetic Materials
Worked example · free
Net field and null point between two parallel currents
- 1At the midpoint each wire is 0.025 m away: B₁ = μ₀(6.0)/(2π × 0.025) = 4.8 × 10⁻⁵ T and B₂ = μ₀(10)/(2π × 0.025) = 8.0 × 10⁻⁵ T.
- 1Between same-direction currents the two fields point in opposite directions, so the net field is 8.0 − 4.8 = 3.2 × 10⁻⁵ T, in the direction of the 10 A wire's field.
- 1Null point: the fields cancel where 6.0/x = 10/(0.050 − x), with x measured from the 6.0 A wire.
- 1Solving: 0.30 − 6.0x = 10x, so x = 0.019 m, about 1.9 cm from the weaker current.
Key terms
- Biot-Savart Law
- The Biot-Savart law gives the magnetic field of a small current element, dB = (μ₀/4π) i ds × r̂ / r², which is summed over a whole circuit.
- Permeability of Free Space
- The permeability of free space μ₀ = 4π × 10⁻⁷ T m/A is the constant that sets the strength of magnetic fields produced by currents in vacuum.
- Ampere's Law
- Ampere's law states that the line integral of B around a closed loop equals μ₀ times the current passing through the loop.
- Amperian Loop
- An Amperian loop is an imaginary closed path chosen so that B is constant and parallel or perpendicular to it, making the line integral easy to evaluate.
- Solenoid
- A solenoid is a long helical coil whose interior field is nearly uniform, B = μ₀ni, where n is the number of turns per metre.
- Ferromagnetic Core
- A ferromagnetic core is a material such as iron whose atomic moments align with an applied field, greatly increasing the field inside a coil.
Magnetic Fields from Currents: Biot-Savart and Ampere FAQ
When should I use Ampere's law instead of Biot-Savart?
Use Ampere's law when the current distribution is symmetric enough that B is constant along a loop you can draw, such as a long wire, a coaxial cable, a solenoid or a toroid. For arcs, rings or finite segments, the symmetry is missing and Biot-Savart is needed.
Why do parallel currents attract?
Each wire sits in the circular field of the other. Applying the force rule F = iL × B to the second wire in the first wire's field gives a force pointing towards the first wire when the currents run the same way, and away from it when they run in opposite directions.
What assumptions does the solenoid derivation need?
A model answer states that the field inside is uniform and parallel to the axis, that the field outside is negligible, and that the rectangular loop encloses N turns each carrying the current i. With those stated, Ampere's law gives B = μ₀ni.
Does the field inside a long solenoid depend on its radius?
No. For an ideal long solenoid the interior field depends only on the number of turns per metre and the current. Near the ends the field weakens and spreads out, so the formula describes the middle of a long coil rather than its ends.
Exam move
Practise both routes to the long-wire field: the Biot-Savart integral with the integral provided, as past papers do, and the two-line Ampere argument. Then run Ampere's method on a deliberately varied set, a thick wire inside and out, a coaxial cable in three regions, a solenoid and a toroid, writing the loop, the line integral and the enclosed current as separate lines each time.
Add direction drills with the grip rule, and finish with force-per-metre problems for parallel wires, stating attraction or repulsion.
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