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Faraday's law of induction is a fundamental principle that describes how a changing magnetic environment can induce an electric field. When a magnetic field through a given area changes over time, it generates an electromotive force (emf) in any nearby conductive path. This is mathematically expressed as \( \text{emf} = -\frac{d\text{Φ}_B}{dt} \), where \( \text{Φ}_B \) is the magnetic flux through the area of interest. This negative sign is a consequence of Lenz's Law, which states that the induced emf will produce a current that opposes the change in magnetic flux that produced it. In the context of the solenoid example, as the current through the solenoid increases, it causes a change in the magnetic flux, which in turn induces an emf along the path, and hence an electric field in the vicinity of the solenoid. This induced electric field is a direct consequence of Faraday's law at work.

The magnetic field inside a solenoid is quite uniform and parallel to the axis of the solenoid, especially near the center. This magnetic field can be calculated using the formula \( B = \text{μ}_0 n i \), with \( \text{μ}_0 \) being the permeability of free space, \( n \) the number of turns per unit length of the solenoid, and \( i \) the current passing through it. This formula is valid for an ideal solenoid where the length is much greater than its diameter. If the current through the solenoid changes, the magnetic field will also change. In our example, as the current rises, the magnetic field inside the solenoid will increase, creating a dynamic environment where Faraday’s law applies to induce an electric field around the solenoid.

The rate of change of current, denoted as \( \frac{di}{dt} \), is a measure of how fast the current through a circuit or component like a solenoid is changing over time. It plays a crucial role in electromagnetic induction because a time-varying current creates a time-varying magnetic field, which is the prerequisite for the induction of an electric field according to Faraday's law. In numerical problems, this rate is often given, as in our solenoid example with a rate of \( 36.0 A/s \). This change in current, when combined with the number of turns and the permeability of free space within Faraday's law, allows us to calculate the induced electric field for different points relative to the solenoid axis.

Permeability of free space, symbolized by \( \text{μ}_0 \), is a constant representing the extent to which a magnetic field can penetrate space. It is one of the fundamental constants of physics, indicating the capability of the vacuum to support the formation of a magnetic field. The value of \( \text{μ}_0 \) is approximately \( 4\text{π} \times 10^{-7} T\text{m/A} \), which plays an integral part in the formulas for the magnetic field in various configurations. In our exercise, the permeability of free space allows us to relate the physical properties of the solenoid—its number of turns per meter and the changing current—to calculate the magnetic field and therefore determine the induced electric field at various points near the solenoid.