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Magnetic flux is a measure of the magnetic field passing through a given area. Imagine it as the number of magnetic lines crossing through a surface. It is represented by the symbol \( \phi \) and is measured in units of Weber (Wb). Magnetic flux depends on two key factors:

• The strength of the magnetic field (B)
• The area (A) of the surface perpendicular to the field

Together, it’s expressed mathematically as \( \phi = B \times A \times \cos(\theta) \), where \( \theta \) is the angle between the magnetic field and the normal (perpendicular) to the surface.In the context of the given exercise, magnetic flux varies with time according to the equation \( \phi = 6t^2 - 5t + 1 \). This equation implies that the flux changes as the time \( t \) changes. The goal in the problem is to find out how this changing flux induces an electromotive force (emf) which results in a current flowing through a circuit.

Ohm's Law is a fundamental principle in electrical circuits, relating the current flow through a conductor to the voltage across it and its resistance. It is expressed by the formula:\[ I = \frac{V}{R} \]where:

• \( I \) is the current in amperes (A)
• \( V \) is the voltage in volts (V)
• \( R \) is the resistance in ohms (\( \Omega \))

In this exercise, following Faraday's Law of Induction, once we have found the induced emf in the circuit as 2 volts, Ohm’s Law helps us determine the current. With a given resistance of 10 ohms, the induced current is found by rearranging the formula: \( I = \frac{\varepsilon}{R} = \frac{2}{10} = 0.2 \) amperes. This means the induced current at \( t = 0.25 \) seconds is 0.2 amperes. Ohm’s Law is invaluable in circuit calculations, providing a straightforward way to relate these three key quantities.

The induced electromotive force, often abbreviated as emf, is a result of changing magnetic fields within a closed loop of wire, according to Faraday's Law of Induction. It is the driving force that causes electrons to move and create an electric current. Mathematically, Faraday's Law is expressed as:\[ \varepsilon = -\frac{d\phi}{dt} \]Here, \( \varepsilon \) represents the induced emf, and \( \frac{d\phi}{dt} \) is the rate of change of magnetic flux. The negative sign is a reflection of Lenz's Law, indicating that the direction of induced emf and current will oppose the change in flux. This negative sign is crucial because it ensures the conservation of energy in electromagnetic processes.In the problem, after differentiating the flux equation \( \phi = 6t^2 - 5t + 1 \), the change in flux \( \frac{d\phi}{dt} \) at \( t = 0.25 \) seconds is calculated as \(-2\). Thus, the induced emf \( \varepsilon = 2 \) volts. The positive value for the emf tells us about the magnitude of the force inducing current around the circuit. Understanding how an emf is induced is essential for comprehending the workings of generators, transformers, and various electrical devices.