A conducting loop with area \(0.15 \mathrm{m}^{2}\) and resistance \(6.0 \Omega\) lies in the \(x\) -y plane. A spatially uniform magnetic field points in the z-direction. The field varies with time according to \(B_{z}=a t^{2}-b\) where \(a=2.0 \mathrm{T} / \mathrm{s}^{2}\) and \(b=8.0 \mathrm{T} .\) Find the loop current (a) at \(t=3.0 \mathrm{s}\) and \((\mathrm{b})\) when \(B_{z}=0.\)

One of the cornerstones of electromagnetism is Faraday's Law, which is key to understanding how electric currents can be induced by changing magnetic fields. It states that an electromotive force (EMF) is induced in a circuit if the magnetic flux through the circuit changes over time. Mathematically, Faraday's Law is expressed as
\( \varepsilon = -\frac{d\Phi}{dt} \),
where \( \varepsilon \) stands for the induced electromotive force and \( \Phi \) represents the magnetic flux. The negative sign, known as Lenz's Law, tells us that the induced EMF creates a current whose own magnetic field opposes the change in magnetic flux that produced it.
In the provided exercise, the magnetic flux through the conducting loop was found by multiplying the magnetic field (which varies over time) by the area of the loop. Using Faraday's Law, by differentiating the product of the magnetic field and the loop area with respect to time, students can find the rate of change of magnetic flux and subsequently the induced EMF. Understanding the relationship between EMF, magnetic flux, and time is essential for mastering the principles of electromagnetism.

Electromotive force, commonly abbreviated as EMF, is the energy provided per charge that passes through a source of electric energy. It is often mistaken for a force; however, it's actually a voltage, and it is measured in volts (V). EMF is the prime mover that causes electric current to flow through a circuit. In the context of Faraday's Law, the EMF is generated as a result of a change in magnetic flux.
To calculate the EMF in our exercise, we employed the rate of change of the magnetic field over time. This produced an expression that, when calculated at a specific time or condition (when \( B_z = 0 \)), gives us the magnitude of the EMF. This is not just theoretical; such principles are fundamental to many technologies, including electric generators and transformers. Students should be aware that while EMF drives the current, it is not the same as the current itself, which is the flow of electric charge.

Ohm's Law is a fundamental principle in the field of electricity which provides a simple relation between voltage, current, and resistance. It states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. The mathematical formula is given by:
\( I = \frac{V}{R} \),
where \( I \) is the current in amperes (A), \( V \) is the voltage in volts (V), and \( R \) is the resistance in ohms (\( \Omega \)). In the exercise, we use Ohm's Law to find the current by dividing the induced EMF by the resistance of the loop. By understanding Ohm's Law, students can deduce how variations in EMF or resistance will affect the current in a circuit. This law is exceedingly practical because it not only applies to simple circuits but is also helpful in analyzing complex electrical networks.