91Ó°ÊÓ

Magnetic flux is a fundamental concept related to Faraday's Law of Induction and is denoted by the symbol \( \Phi_B \). It represents the amount of magnetic field passing through a given area. In practical terms, it helps us understand how magnetic fields interact with physical objects, such as coils and loops. For a uniform magnetic field, magnetic flux through a surface can be calculated with the formula: \( \Phi_B = \int \mathbf{B} \cdot d\mathbf{A} \), where:

• \( \mathbf{B} \) is the magnetic field vector.
• \( d\mathbf{A} \) is a differential area vector perpendicular to the surface.

In our problem, the magnetic field is sinusoidal inside a solenoid, described by \( \mathbf{B}(t) = B_0 \cos(\omega t) \hat{z} \). With a circular loop inside the solenoid, the area \( \frac{a^2 \pi}{4} \) is constant, allowing us to express the magnetic flux as \( \Phi_B(t) = B_0 \pi \left( \frac{a}{2} \right)^2 \cos(\omega t) \). This time-varying flux is key for calculating the electromotive force (emf).

Electromotive force, commonly referred to as emf, is the energy provided per charge by a source of electrical energy. According to Faraday's Law of Induction, the emf induced in a circuit is equal to the negative rate of change of magnetic flux through the circuit. This can be mathematically expressed as \( \text{emf} = - \frac{d\Phi_B}{dt} \). To determine the emf in our exercise:

• We differentiate the expression for magnetic flux \( \Phi_B = B_0 \pi \left( \frac{a}{2} \right)^2 \cos(\omega t) \) with respect to time \( t \).
• Using the chain rule, the derivative of \( \cos(\omega t) \) with respect to time is \( -\omega \sin(\omega t) \).
• This gives us: \( \text{emf} = B_0 \pi \left( \frac{a}{2} \right)^2 \omega \sin(\omega t) \).

Through this process, we convert a time-varying magnetic flux into an induced emf, which can drive current through a loop or circuit.

Ohm’s Law is a cornerstone of basic electronics and physics, establishing the relation between voltage (\text{emf), current \((I)\), and resistance \((R)\). It is represented by the equation: \( I = \frac{\text{emf}}{R} \). In simple terms, it tells us that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor.

• In the context of our exercise, we've calculated the induced emf due to the changing magnetic flux.
• By applying Ohm’s Law, we determine the induced current in the loop: \( I(t) = \frac{B_0 \pi \left( \frac{a}{2} \right)^2 \omega \sin(\omega t)}{R} \).
• Simplifying further, \( I(t) = \frac{B_0 \pi a^2 \omega}{4R} \sin(\omega t) \), shows how resistance, the geometry of the loop, and the magnetic field dynamics influence the current.

This highlights Ohm's Law's utility in connecting the concepts of electromotive force and magnetic flux to tangible electrical characteristics like current and resistance.