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Faraday's Law is one of the fundamental principles explaining how electric currents can be generated by changing magnetic fields. It is at the heart of electromagnetic induction — a key process in many electrical systems such as transformers and generators.
When the magnetic environment of a circuit changes, it influences the circuit and an electromotive force (emf) is induced. The core idea is that a changing magnetic field over time will produce electricity. This is expressed mathematically by the formula:
\[ \varepsilon = - \frac{d\phi}{dt} \]
where \( \varepsilon \) is the induced emf, and \( \frac{d\phi}{dt} \) is the rate of change of the magnetic flux \( \phi \) with respect to time.
The negative sign is crucial because it indicates the direction of the induced emf which opposes the change in flux, as established by Lenz's Law. This means that the circuit will resist changes in its magnetic environment.

Magnetic Flux quantifies the total magnetic field passing through a given area. It is a useful way to understand and calculate the influence of a magnetic field in electromagnetism.
Magnetic flux is denoted by the symbol \( \phi \) and is measured in Weber (Wb). It can be given by the formula:
\[ \phi = B \cdot A \cdot \cos(\theta) \]
where:

• \( B \) is the magnetic field strength (in Tesla)
• \( A \) is the area through which the field lines pass (in square meters)
• \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface

In practical terms, whenever you change the magnetic field strength, the area, or the orientation of the circuit, you change the magnetic flux. As the exercise shows, if the magnetic flux changes with time, this will induce an emf according to Faraday's Law.

Differentiation is a mathematical technique used to determine the rate at which a quantity changes with respect to another quantity. In the context of the exercise, it is essential for calculating the rate of change of magnetic flux.
To find the induced emf, you differentiate the expression for magnetic flux, which is given as \( \phi(t) = 3t^2 + 4t + 9 \).
The differentiation process involves applying basic calculus rules:
\[ \frac{d\phi}{dt} = \frac{d}{dt}(3t^2 + 4t + 9) = 6t + 4 \]
This derivative \( 6t + 4 \) represents how the flux changes at any given moment \( t \). Evaluating this at \( t = 2 \) gives you \( 16 \). The differentiation allows you to pinpoint the exact rate of flux change, which directly leads to calculating the induced emf as per Faraday's Law.