The number of turns in the coil is directly proportional to the induced emf. More turns mean more pathways for the current, leading to an increase in the total induced emf, according to Faraday’s law where ε = -N(ΔΦ/Δt). Each turn of the coil experiences an induced emf, and having multiple turns sums up the individual emfs, leading to a higher total induced emf. In applications like generators, increasing the number of turns is a practical approach to amplify the output voltage, enhancing the efficiency of energy conversion and transmission.

Coils in this context are typically made of conductive materials with low resistance to allow efficient flow of induced current. Copper is a common choice due to its high electrical conductivity and low resistivity, ensuring that the induced emf results in a substantial current. Additionally, copper's mechanical strength and ductility make it a suitable material to withstand the physical stresses involved in rapid rotation. The coil’s material is a key factor influencing the efficiency and effectiveness of energy conversion from mechanical form to electrical form in applications like generators.

In generators, the sinusoidal varying emf induced due to the rotation of coils in a magnetic field results in alternating current (AC) output. However, many applications require direct current (DC). To convert AC to DC, a rectifier is used, which allows current to flow in only one direction, thus converting the alternating current into a unidirectional current. Additionally, to smooth out the ripples in the rectified output and make it more constant, filters like capacitors or inductors are often added to the circuit. They store energy during the peak of the wave and release it during the trough, providing a more consistent DC output.

The size of the coil significantly influences the induced emf. A larger coil encapsulates more magnetic field lines, increasing the magnetic flux. According to Faraday’s law, the induced emf is proportional to the rate of change of magnetic flux. Therefore, a larger coil, having more area, will have a larger change in magnetic flux for a given change in the angle between the magnetic field and the normal to the coil. This results in a higher induced emf. However, it's also crucial to consider other parameters like the coil’s resistance which increases with the coil size, potentially impacting the current induced.

Yes, the shape of the sinusoidal wave of induced emf can be modified using various electrical components and techniques. In many applications, especially in power electronics, it’s often desirable to modify the waveform to control the power delivered to a load. Techniques like pulse-width modulation (PWM) can be employed, where the waveform is modified by varying the duty cycle of a pulse wave to approximate a sinusoidal wave. Transformers and inductors can also be used to smooth out the waveform, reducing harmonics and producing a wave that’s closer to a pure sinusoidal form for specific application needs.