Difficulty: Intermediate–Advanced | Prerequisites: Lorentz force, cross products, integration basics, symmetry arguments.
Once you understand the force on a single moving charge, scaling up to wires carrying current is straightforward: a current is simply many charges moving together. This gives you the force on a wire, the torque on a current loop, and the magnetic dipole moment. On the flip side, currents also create magnetic fields, and the two main tools for calculating those fields are the Biot-Savart Law (general, works everywhere) and Ampere's Law (simpler, but only practical when there is high symmetry). Together, these ideas let you both predict how conductors respond to external fields and calculate the fields that conductors produce.
A current-carrying wire in a magnetic field feels a force F = IL × B. A current loop acts like a magnetic dipole with moment μ = NIA, and it experiences a torque τ = μ × B that tries to align it with the field. The Biot-Savart Law gives the magnetic field from any current element, while Ampere's Law provides a shortcut for symmetric geometries like infinite wires, solenoids, and current sheets.
Force on a current-carrying wire
F = IL × B, where I is the current, L is the length vector (direction of current flow), and B is the external magnetic field. Think of it as the Lorentz force summed over all the moving charges in the wire.
Magnetic dipole moment (μ)
For a current loop: μ = NIA, where N is the number of loops, I is the current, and A is the area vector (direction from the right-hand rule, curling fingers in the direction of current). It quantifies how strongly a loop interacts with an external field.
Torque on a current loop
τ = μ × B. The torque always acts to align the dipole moment with the external field.
Magnetic potential energy
U = −μ · B = −μBcosθ. Energy is minimised when μ is aligned with B (θ = 0) and maximised when they are anti-aligned (θ = π).
Biot-Savart Law
The fundamental law for finding the magnetic field due to a small current element: dB = (μ₀/4π)(I ds × r̂)/r². Think of it as the magnetic equivalent of Coulomb's Law, but for currents instead of charges.
Ampere's Law
∮ B · dl = μ₀ I_enclosed. The line integral of B around any closed loop (Amperian loop) equals μ₀ times the total current passing through the loop. It is the magnetic analogue of Gauss's Law: powerful when symmetry is present, awkward when it is not.
Solenoid
A tightly wound coil of wire. Inside an ideal solenoid, the magnetic field is uniform and parallel to the axis: B = μ₀nI, where n = N/L is the number of turns per unit length.
Permeability of free space (μ₀)
μ₀ = 4π × 10⁻⁷ T·m/A. The fundamental constant that sets the strength of magnetic interactions in vacuum.
Start from the single-particle Lorentz force: F = qv × B.
For a collection of charges in a wire: F = qNv_avg × B, where N is the total number of charge carriers.
N = nAL (n = charge carrier density, A = cross-sectional area, L = length).
Current: I = qnAv_avg.
This simplifies to: F = IL × B
The direction of L follows the direction of conventional current.
The magnitude is F = BILsinθ, where θ is the angle between the wire and the field.
τ = R × F (radius vector crossed with force vector, standard torque definition).
For a current loop with dipole moment μ = NIA: τ = μ × B
The torque drives the loop to rotate so that μ aligns with B.
At equilibrium (μ parallel to B), τ = 0 and energy is at a minimum.
Work done by torque: W = ∫τ · dθ
Since τ = μBsinθ, integrating gives: W = μBcosθ = μ · B
Potential energy: U = −μ · B = −μBcosθ
U is lowest (most stable) when μ and B are parallel (θ = 0).
U is highest (least stable) when μ and B are anti-parallel (θ = π).
Each wire creates a magnetic field that acts on the other wire.
Using F = IL × B, where B is the field from the other wire:
Like currents (same direction) attract.
Opposite currents (opposite direction) repel.
Similarly, current loops oriented the same way attract, and opposite loops repel.
dB = (μ₀/4π) (I ds × r̂) / r²
ds is a small element of the wire, in the direction of current.
r̂ is the unit vector pointing from the current element to the field point.
r is the distance from the current element to the field point.
To find the total field, integrate over the entire current distribution.
Standard results from Biot-Savart:
Infinite straight wire: B = μ₀I/(2πR), where R is the perpendicular distance from the wire. Field lines are circles centred on the wire.
Current loop on axis: B = (μ₀IR²) / [2(R² + z²)^(3/2)], where R is the loop radius and z is the distance along the axis from the centre.
∮ B · dl = μ₀ I_enclosed
Choose a closed path (Amperian loop) that exploits the symmetry.
B must be constant along the path (or zero, or perpendicular to dl) for the integral to simplify.
Only the current passing through the enclosed area of the loop contributes.
This gives the same results as Biot-Savart for symmetric cases, but much more quickly.
Assumptions: uniform field B inside, zero field outside, field parallel to the solenoid axis.
Rectangular Amperian loop with one side of length L inside the solenoid:
BL = μ₀nLI, where n = N/L (turns per unit length).
B = μ₀nI inside the solenoid.
The field is independent of position inside an ideal (infinite) solenoid.
A planar sheet with wire density n (wires per unit length).
Rectangular Amperian loop straddling the sheet:
2BL = μ₀nLI
B = μ₀nI/2 on each side of the sheet.
For an infinitely long cylindrical shell carrying uniformly distributed current I, with inner radius a and outer radius b:
r < a (inside the hollow): No enclosed current → B = 0.
r > b (outside the shell): All current enclosed → B = μ₀I/(2πr), identical to a long straight wire.
a < r < b (within the shell): Only the current inside radius r is enclosed.
Current density: j = I/[π(b² − a²)]
Enclosed current: I_enc = I(r² − a²)/(b² − a²)
B = (μ₀I)/(2πr) · (r² − a²)/(b² − a²)
B grows from 0 at r = a, reaches a maximum, then falls off as 1/r outside.
Quantity | Formula |
|---|---|
Force on a wire | F = IL × B, magnitude = BILsinθ |
Magnetic dipole moment | μ = NIA |
Torque on a dipole | τ = μ × B |
Magnetic potential energy | U = −μ · B = −μBcosθ |
Biot-Savart Law | dB = (μ₀/4π)(I ds × r̂)/r² |
B from infinite wire | B = μ₀I/(2πR) |
B from loop on axis | B = μ₀IR²/[2(R² + z²)^(3/2)] |
Ampere's Law | ∮ B · dl = μ₀I_enclosed |
B inside solenoid | B = μ₀nI |
B from current sheet | B = μ₀nI/2 |
Electric motors work by running current through a coil in a magnetic field, producing torque that rotates a shaft. MRI machines use solenoid-like superconducting coils to generate the strong, uniform magnetic fields needed to image the body. The force between parallel current-carrying wires was historically used to define the ampere (before the 2019 SI redefinition).
Students confuse the direction of the area vector A in the dipole moment with the plane of the loop. The area vector is perpendicular to the loop, determined by curling the right-hand fingers in the direction of current flow.
Ampere's Law works for any closed path, but it only simplifies the calculation when the symmetry is right. Using it for an irregular current distribution is technically correct but practically useless.
The field outside a solenoid is not exactly zero; it is approximately zero for an ideal, infinitely long solenoid. Real solenoids have fringe fields at the ends.
Like currents attract. Students sometimes guess that because like charges repel, like currents should also repel. The magnetic interaction is different from the electrostatic one.
⚠️ The cylindrical shell example (splitting into three regions with Ampere's Law) is a classic exam problem. Know how to set up each region.
⚠️ Be able to derive B inside a solenoid from Ampere's Law, not just state the result.
⚠️ Torque and potential energy problems for magnetic dipoles are frequently tested: know the difference between stable and unstable equilibrium.
⚠️ The Biot-Savart Law for an infinite wire and for a loop on axis are standard results you should memorise.
1. Fill in the blank: The magnetic dipole moment of a 200-turn coil carrying 3 A with area 0.01 m² is μ = ______.
A: μ = NIA = 200 × 3 × 0.01 = 6 A·m².
2. True or false: Two parallel wires carrying currents in the same direction repel each other.
A: False. Like currents attract.
3. Fill in the blank: Inside an ideal solenoid with 1000 turns per metre carrying 2 A, the magnetic field is B = ______.
A: B = μ₀nI = (4π × 10⁻⁷)(1000)(2) = 2.51 × 10⁻³ T ≈ 2.5 mT.
4. True or false: Ampere's Law can only be applied to straight wires.
A: False. It applies to any current distribution, but it only simplifies calculations when there is sufficient symmetry.
Q: A square current loop (side length 0.1 m, current 5 A, 1 turn) sits in a uniform 0.2 T magnetic field. What is the maximum torque on the loop?
A: μ = IA = 5 × (0.1)² = 0.05 A·m². Maximum torque (when μ ⊥ B): τ = μB = 0.05 × 0.2 = 0.01 N·m.
Q: At what angle between μ and B is the potential energy zero?
A: U = −μBcosθ = 0 when cosθ = 0, so θ = 90° (π/2).
Q: Using Ampere's Law, find B at distance r from an infinite straight wire carrying current I.
A: Choose a circular Amperian loop of radius r centred on the wire. By symmetry, B is constant and parallel to dl everywhere on the loop. ∮B·dl = B(2πr) = μ₀I. Therefore B = μ₀I/(2πr).
Q: A cylindrical shell (inner radius 2 cm, outer radius 5 cm) carries 10 A uniformly distributed. What is B at r = 3 cm?
A: Region a < r < b. I_enc = I(r² − a²)/(b² − a²) = 10(0.03² − 0.02²)/(0.05² − 0.02²) = 10(0.0005)/(0.0021) = 2.38 A. B = μ₀I_enc/(2πr) = (4π × 10⁻⁷)(2.38)/(2π × 0.03) = 1.59 × 10⁻⁵ T ≈ 15.9 μT.
The force on a wire (F = IL × B) is a direct generalisation of the Lorentz force on a single charge. The Biot-Savart Law and Ampere's Law are the tools you need for calculating magnetic fields, which feed into Faraday's Law and motional EMF (next topics). The solenoid result is essential for understanding inductors in RL and RLC circuits later in the course.
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