Unit 16: Magnetic Field

F = I (L × B)

Magnitude:

F = B I L sin θ

Where θ is the angle between the conductor’s length vector (direction of current) and the magnetic field.

Applications include:

• Electric motors – torque on a current loop in a magnetic field produces rotation.
• Mass spectrometers – charged particles are deflected by magnetic forces to separate isotopes.
• Magnetic levitation (maglev) – repulsive or attractive forces between currents enable frictionless transport.

3. Torque on a Coil and Moving Coil Galvanometer

Torque on a Rectangular Coil

Consider a rectangular coil of N turns, each turn having area A, carrying a current I, placed in a uniform magnetic field B. The normal to the coil makes an angle θ with the field. The torque tending to rotate the coil is:

τ = N I A B sin θ

Derivation (brief): Each side of length l experiences a force F = B I l. The two forces produce a couple with lever arm w/2 (where w is the width). Summing over N turns gives the expression above.

The torque is maximum when the coil’s plane is parallel to the field (θ = 90°) and zero when the coil is aligned with the field (θ = 0°).

Moving Coil Galvanometer

A moving coil galvanometer measures small electric currents by detecting the torque on a coil suspended in a radial magnetic field. Its main parts are:

• A lightweight rectangular coil wound on a non‑magnetic former.
• A permanent magnet that provides a radial magnetic field (so that the plane of the coil is always parallel to the field, making sin θ ≈ 1).
• A fine phosphor‑bronze strip that provides a restoring torque proportional to the angular deflection.
• A pointer attached to the coil that moves over a calibrated scale.

At equilibrium, magnetic torque equals restoring torque:

N I A B = k θ

Where k is the torsion constant of the suspension.

Thus the deflection θ is directly proportional to the current I, giving a linear scale. By adding a shunt resistor the device can be used as an ammeter; by adding a high‑series resistance it becomes a voltmeter.

4. Hall Effect

When a current‑carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current, a voltage develops across the specimen in a direction perpendicular to both the current and the field. This transverse voltage is the Hall voltage.

Hall Voltage Formula

For a slab of thickness b (dimension along the magnetic field), width w (along which the Hall voltage is measured), carrying current I, the Hall voltage is:

V_H = \frac{B I}{n q b}

Where:

• B – magnetic field strength (T)
• I – current (A)
• n – charge carrier density (carriers per m³)
• q – magnitude of the charge of the carrier (C)
• b – thickness of the specimen in the direction of B (m)

The sign of V_H reveals whether the dominant carriers are positive (holes) or negative (electrons).

Applications

• Hall sensors – used as proximity switches, speed detectors, and current sensors.
• Magnetometers – measure magnetic field strength with high sensitivity.
• Characterisation of semiconductors – determines carrier type and density, essential for device design.

5. Biot‑Savart Law

The Biot‑Savart law gives the magnetic field dB produced at a point by an infinitesimal current element I d\mathbf{l}.

Mathematical Form

d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{\hat{r}}}{r^2}

Where:

• \mu_0 = 4\pi × 10^{-7} T·m/A – permeability of free space.
• d\mathbf{l} – vector length of the current element, direction of current.
• \mathbf{\hat{r}} – unit vector from the element to the field point.
• r – distance from the element to the point.

The total field is obtained by integrating over the entire current distribution: \mathbf{B} = \int d\mathbf{B}.

Applications

1. Magnetic field at the centre of a circular loop (radius R):
   B = \frac{\mu_0 I}{2R} (direction given by right‑hand rule).
2. Field on the axis of a circular loop (distance x from centre):
   B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}.
3. Long straight wire:
   B = \frac{\mu_0 I}{2\pi r}, where r is the perpendicular distance from the wire.
4. Solenoid (ideal, infinite length):
   B = \mu_0 n I (inside), where n is turns per unit length.

6. Ampère’s Law

Ampère’s law relates the line integral of the magnetic field around a closed loop to the net current passing through the loop.

Integral Form

\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}

Where I_{\text{enc}} is the total current enclosed by the Amperian loop.

The law is especially useful for highly symmetric current distributions.

Applications

• Long straight wire: Choose a circular Amperian loop of radius r centred on the wire. Symmetry gives B constant on the loop, leading to B = \frac{\mu_0 I}{2\pi r}.
• Solenoid: For an ideal solenoid, the field inside is uniform and parallel to the axis; outside it is negligible. Applying Ampère’s law to a rectangular loop that goes inside and outside yields B = \mu_0 n I.
• Toroid: A toroidal coil of N turns carrying current I produces a field confined within the core:
  B = \frac{\mu_0 N I}{2\pi r}, where r is the radial distance from the centre of the toroid.

7. Force Between Parallel Conductors

Two long, straight, parallel conductors separated by a distance d and carrying currents I_1 and I_2 exert a magnetic force on each other. The force per unit length is:

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

If the currents are in the same direction, the force is attractive; if opposite, it is repulsive.

This expression forms the basis for the SI definition of the ampere:

One ampere is the constant current which, if maintained in two straight parallel conductors of infinite length, negligible circular cross‑section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10^{-7} newton per metre of length.

Summary

This chapter has provided a detailed treatment of magnetostatics:

• Magnetic field lines give a visual picture; magnetic flux quantifies the field through a surface.
• The Lorentz force governs the interaction of magnetic fields with moving charges and currents, leading to the force on a conductor F = BIL sinθ.
• A current loop experiences torque τ = NIAB sinθ, the principle behind moving coil galvanometers.
• The Hall effect reveals charge carrier type and density via V_H = BI/(nqb).
• The Biot‑Savart law allows calculation of magnetic fields from arbitrary current distributions; its applications include loops, straight wires, and solenoids.
• Ampère’s law offers a powerful shortcut for symmetric configurations, reproducing the same results for wires, solenoids, and toroids.
• Finally, the force between parallel conductors leads to the operational definition of the ampere and underpins the design of many electromagnetic devices.

Together, these concepts form the foundation for understanding electromagnetism and its technological applications.