Loudspeakers

A loudspeaker contains a cylindrical permanent magnet (usually made of ferrite or neodymium) with a carefully shaped pole piece. The geometry creates a strong, radially directed magnetic field $\vec{B}$ in the narrow annular gap where the voice coil sits. Typical field strengths range from 0.5 T to 2 T in high-quality speakers. Because the field is radial, it is always perpendicular to the wire in the coil regardless of the coil's position, which maximises the force at every point in the cycle.

The voice coil is a tightly wound coil of thin insulated copper wire attached to the neck of the cone and positioned in the magnetic gap. An audio amplifier sends a time-varying current $I(t)$ through this coil. The current waveform is an electrical analogue of the original sound: its frequency matches the frequency of the desired tone and its amplitude determines the loudness. For a standard 8 Ω speaker, a signal voltage of a few volts produces currents of hundreds of milliamperes.

Each straight segment of wire carrying current $I$ in magnetic field $B$ experiences a force given by $F = BIL$, where $L$ is the total effective length of wire in the gap. Because all segments of the circular coil experience forces in the same axial direction (either forward or backward, depending on current direction), the net force accelerates the coil axially. When the current reverses, so does the force, making the coil oscillate at the frequency of the audio signal.

The voice coil is mechanically coupled to a large, lightweight cone — typically made of paper, aluminium, or polypropylene. The cone acts as a piston: as it accelerates back and forth, it displaces the surrounding air, creating alternating regions of compression (high pressure) and rarefaction (low pressure). These pressure waves propagate outward at the speed of sound (~343 m/s in air at 20 °C) and are detected by our ears as sound. The cone's large surface area is essential for efficient coupling of energy into air, because the acoustic impedance mismatch between a small coil and air would otherwise be severe.

As the coil moves, it acts as a generator: the changing flux through the coil induces a back-EMF $\varepsilon = -N\frac{d\Phi}{dt} = -BLv$, where $v$ is the coil velocity and $L$ is the effective wire length in the field. This back-EMF opposes the driving current (Lenz's law), providing electromagnetic damping that helps control the cone's motion and prevents unwanted resonance — an effect called motional damping or electrical braking.

F \approx 1.49 \text{ N}, \quad a = \frac{F}{m} \approx 186 \text{ m s}^{-2}

• Magnetic Force on a Current-Carrying Conductor — The equation F = BIL is the direct physical basis for the loudspeaker voice coil force. Understanding how and why a current-carrying wire in a field experiences a force is essential before analysing any electromechanical device.
• Electromagnetic Induction and Lenz's Law — The back-EMF generated by the moving coil is an induction effect described by Faraday's and Lenz's laws. This back-EMF provides damping and is crucial for understanding the speaker's frequency response and impedance.
• Permanent Magnets and Magnetic Materials — The strength and geometry of the permanent magnet determines the field B in the gap. Understanding ferromagnetic materials and how permanent magnets work explains why speaker manufacturers choose neodymium over ferrite for high-performance applications.
• Sound Waves and Pressure Variation — The cone's vibration produces longitudinal pressure waves in air. Connecting the mechanical motion of the cone to the physics of sound propagation completes the energy conversion chain from electrical signal to audible sound.