An 8 -turn coil has square loops measuring \(0.200 \mathrm{~m}\) along a side and a resistance of \(3.00 \Omega\). It is placed in a magnetic field that makes an angle of \(40.0^{\circ}\) with the plane of each loop. The magnitude of this field varies with time according to \(B=1.50 t^{3}\), where \(t\) is measured in seconds and \(B\) in teslas. What is the induced current in the coil at \(t=2.00 \mathrm{~s} ?\)

Understanding magnetic flux is crucial to comprehending how induced currents in coils work. Magnetic flux, denoted by the Greek letter \( \Phi \), refers to the amount of magnetic field that passes through a given area. It's the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field direction and the normal to the surface (\( \cos(\theta) \) for flat surfaces). In mathematical terms, \( \Phi = B \times A \times \cos(\theta) \), where \( B \) is the magnetic field, \( A \) is the area, and \( \theta \) is the angle between the field lines and the perpendicular to the surface.

Now, when it comes to a coil with multiple turns, we must consider each loop. The total flux through the coil is essentially the sum of the flux through each turn. Therefore, if you have a coil with \( n \) turns, the total magnetic flux would be \( n \times \Phi \) for each individual loop. This concept directly informs Faraday's law, which relates magnetic flux to induced electromotive force (emf).

Faraday's law of electromagnetic induction is a cornerstone of electromagnetism, defining how a voltage or emf is induced in a circuit due to changes in magnetic flux. Faraday's law states that the induced emf in any closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. The formula is beautifully simple: \( \epsilon = -n \frac{d\Phi}{dt} \) where \( \epsilon \) is the induced emf, \( n \) is the number of turns of the coil, and \( \frac{d\Phi}{dt} \) represents the rate of change of magnetic flux over time.

When the change in magnetic flux is positive, the negative sign in Faraday's law tells us that the induced current creates a magnetic field opposing the original change, a marvelously self-regulating system consistent with Lenz's law. The usefulness of Faraday's law is immense; it forms the working principle behind many electrical machines and transformers.

Lenz's law is a fundamental principle that gives us the direction of the induced current resulting from any change in magnetic flux. Lenz's law states that the direction of any induced current is such that it'll oppose the change in magnetic flux that produced it. This law is essentially a statement of energy conservation and aligns with the negative sign in Faraday's law of electromagnetic induction.

When studying induced currents, Lenz's law is integral to understanding why currents behave as they do. In our coil example, when the magnetic field through the coil increases, the induced current generates a magnetic field that attempts to reduce the increase. Conversely, if the flux decreases, the induced current's magnetic field works to maintain it. Lenz's law helps us predict the effects of electromagnetic induction in various technology, including generators and inductive sensors.

A time-dependent magnetic field is one that changes in magnitude or direction with time. Such variation is essential for electromagnetic induction to occur, according to Faraday's law. Whether the field increases, decreases or rotates with time, any change is capable of inducing a voltage in a neighboring conductor.

For instance, a magnetic field increasing linearly with time — such as the one described by the formula \( B = 1.50t^3 \) — will induce an emf whose magnitude is proportional to the rate of change of the field. The faster the field changes, the greater the induced emf and, consequently, the current in a circuit with a given resistance. Understanding time-dependent fields enables us to engineer devices for converting kinetic energy into electrical energy, like in the case of dynamos and alternators used in power generation.