Magnetic Effects of Current and Magnetism

Biot-Savart law: the magnetic field dB at a point due to a current element I·dL is

dB = (μ₀/4π) · (I dL × r̂) / r²

where r is distance from element to point, r̂ is unit vector, μ₀/4π = 10⁻⁷ T·m/A.

Direction by right-hand rule: from dL toward r̂.

This is the magnetic analog of Coulomb's law: superposition principle applies; integrate over all current elements to find total B.

Standard field configurations (derived from Biot-Savart):

1. Straight infinite wire at distance d:

B = μ₀I / (2π d)

Field forms concentric circles around the wire (right-hand grip rule).

2. Straight wire of finite length:

B = (μ₀I / 4πd)(sin θ₁ + sin θ₂)

where θ₁ and θ₂ are angles from the perpendicular at the foot to the two ends.

3. Circular loop of radius R, at center:

B = μ₀I / (2R)

Direction: perpendicular to the plane, given by right-hand grip rule on current direction.

4. Circular loop, on axis at distance x from center:

B = μ₀ I R² / [2(R² + x²)^(3/2)]

At x = 0, reduces to formula 3.

5. Solenoid (long, n turns per metre):

Inside: B = μ₀ n I (uniform).
Outside: B ≈ 0.

6. Toroid (donut-shaped solenoid, N total turns, mean radius r):

Inside the core: B = μ₀ N I / (2π r). Outside: 0.

Ampere's circuital law:

∮ B · dL = μ₀ I_enclosed

The line integral of B around a closed loop equals μ₀ times the current threading the loop.

Use Ampere's law when symmetry lets you choose a path where B is constant or zero. Much faster than Biot-Savart for symmetric configurations.

Magnetic field of a moving point charge:

B = (μ₀/4π) · q (v × r̂) / r²

This is the relativistic counterpart of Coulomb's law for a point charge moving with velocity v. (Magnetic field is essentially relativity + electricity.)

Force between two parallel wires:

For two parallel wires carrying currents I₁ and I₂, separated by distance d:

F/L = μ₀ I₁ I₂ / (2π d)

• Same direction → attractive.
• Opposite direction → repulsive.

This is the definition of the ampere (SI): 1 A flowing in two parallel wires 1 m apart produces a force of 2 × 10⁻⁷ N/m.

Worked example. Find B at the center of a circular loop of radius 10 cm carrying 5 A.

B = μ₀I / (2R) = (4π × 10⁻⁷ × 5) / (2 × 0.1) = π × 10⁻⁵ ≈ 3.14 × 10⁻⁵ T = 31.4 μT.

(Roughly comparable to Earth's magnetic field, ~50 μT.)

A charge q moving with velocity v in a magnetic field B experiences:

F = q(v × B)

(In a combined E and B field, total Lorentz force is F = q(E + v × B).)

Magnitude: F = qvB sin θ, where θ is the angle between v and B.

Direction (right-hand rule): point fingers along v, curl toward B. Thumb gives direction of v × B. Reverse for negative charges.

Three special cases:

1. v parallel to B (θ = 0° or 180°): F = 0. Charge continues in straight line.

2. v perpendicular to B (θ = 90°): F = qvB, perpendicular to motion. Force does no work (F · v = 0). Charge moves in a circle of radius:

r = mv / (qB)

with period T = 2πm/(qB), independent of speed. This is cyclotron motion.

3. v at arbitrary angle to B: decompose v into v∥ (parallel) and v⊥ (perpendicular). v∥ unchanged; v⊥ produces circular motion. Net trajectory: helix.

Force on a current-carrying wire: for a wire of length L carrying current I in field B:

F = IL × B, magnitude F = BIL sin θ.

This is how electric motors work — current in loop placed in B-field experiences torque τ = NIAB sin θ that rotates the loop.

Worked example. An electron (q = 1.6×10⁻¹⁹ C, m = 9.1×10⁻³¹ kg) moves at 10⁶ m/s perpendicular to a 0.01 T field. Find the radius of its circular motion.

r = mv/(qB) = (9.1×10⁻³¹ × 10⁶) / (1.6×10⁻¹⁹ × 0.01) = 9.1×10⁻²⁵ / 1.6×10⁻²¹ = 5.7×10⁻⁴ m ≈ 0.57 mm.

Cyclotron motion is the operating principle of mass spectrometers (separating ions by m/q) and particle accelerators (cyclotron, synchrotron).