Read the following passage and answer the accompanying questions - Leaving Cert Physics - Question 11 - 2006

The frequency of vibration of a stretched string is directly proportional to the square root of the tension. Mathematically, this can be expressed as: $f \\propto \\sqrt{T}$ where $f$ is the frequency and $T$ is the tension.

The speed of sound in a stretched string can be calculated using the formula: $v = f imes \lambda$ where $f$ is the frequency and $\lambda$ is the wavelength. For a string fixed at both ends, the fundamental frequency has a wavelength of: $\lambda = 2L$ where $L$ is the length of the string. Given that the length is 0.8 m: $\lambda = 2 \times 0.8 = 1.6 m$ Thus, $v = 400 \, ext{Hz} \times 1.6 \, ext{m} = 640 \, ext{m/s}.$

The strings in the electric guitar are made of steel to ensure that they can be magnetized. This property allows the strings to interact effectively with the magnets positioned under them, thus generating the electromagnetic induction necessary to produce sound.

Magnetic flux ($\Phi$) is defined as the product of the magnetic field ($B$) passing through a given area ($A$) and the cosine of the angle ($\theta$) between the magnetic field lines and the normal to the surface. It can be expressed mathematically as: $\Phi = B \cdot A \cdot \cos(\theta)$

The current produced in the coil of an electric guitar varies due to changes in the magnetic flux cutting through the coil, which happens when the guitar strings vibrate. The varying magnetic field induces an electromotive force (emf) in the coil, leading to fluctuations in the current.

Increasing the number of turns in a coil generally enhances the sound produced by the electric guitar, as it increases the induced emf. This means a louder or greater sound intensity can be achieved, leading to a richer audio experience.

The induced emf ($E$) in a coil is given by Faraday's law of electromagnetic induction, which is: $E = -N \frac{\Delta \Phi}{\Delta t}$ where $N$ is the number of turns, $\Delta \Phi$ is the change in magnetic flux, and $\Delta t$ is the time interval. For 5000 turns, with a change in flux of $8 \times 10^{-4}$ Wb over 0.1 s: $E = -5000 \times \frac{8 \times 10^{-4}}{0.1} = -40 \, ext{V}$ (The negative sign indicates the direction of the induced emf according to Lenz's law.)