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Induced voltage is a crucial concept in understanding how electrical circuits with inductors operate. When we talk about induced voltage, we are referring to the potential difference that is created across an inductor due to its interaction with changing current. In essence, inductors resist changes in the current flowing through them. Thus, when the current changes, whether by increasing or decreasing, an electromotive force (emf) is induced. This is known as induced voltage.

The key to finding the voltage in our exercise is using the formula for the voltage across an inductor:

• The voltage across a single inductor is given by: \( v = L \cdot \frac{di}{dt} \), where \( L \) is inductance and \( \frac{di}{dt} \) is the rate of change of current with respect to time.
• In circuits with mutual inductance, the voltage induced in one coil also depends on the rate of change of current in the other coil. This is represented by: \( v_1 = L_1 \cdot \frac{di_1}{dt} + M \cdot \frac{di_2}{dt} \) and \( v_2 = L_2 \cdot \frac{di_2}{dt} + M \cdot \frac{di_1}{dt} \), where \( M \) is mutual inductance.

Understanding induced voltage helps to predict the behavior of circuits in dynamics and is fundamental in designing transformers and other electromechanical devices.

Coil inductance is an intrinsic property of an inductor that quantifies its ability to resist changes in current. It depends on factors like the number of wire loops (turns) in the coil, the core material, and the coil's dimensions. Inductance is measured in henries (H).

When dealing with multiple coils, we also need to consider mutual inductance, especially if they are closely positioned like in our exercise:

• **Self-Inductance (L):** It is the property of a coil that resists the change in current flowing through it. Every inductor has its own self-inductance defined by \( L_1 \), \( L_2 \), etc.
• **Mutual Inductance (M):** This occurs when two inductors are placed such that the magnetic field of one affects the other, like the exercise where two coils affect each other's output. The changing magnetic field in one coil induces a voltage in the nearby coil, and vice versa. The mutual inductance is determined by layout and the closeness of the coils.

In practical applications, understanding mutual inductance is vital for the design and function of transformers and inductive coupling circuits, as it can either enhance or limit their performance depending on configuration.

The rate at which current changes over time is crucial in understanding how voltages are induced in inductors. This rate is expressed mathematically as a derivative, noted as \( \frac{di}{dt} \). In our exercise, the precise calculation of these derivatives is key in determining the induced voltages across the coils.

Taking derivatives means calculating the slope of the function at a given point, indicating how rapidly the current is changing:

• For a cosine function like \( i_1 = \cos(377t) \), the derivative is \( \frac{di_1}{dt} = -377 \sin(377t) \).
• Similarly, for \( i_2 = 0.5 \cos(377t) \), its derivative is \( \frac{di_2}{dt} = -188.5 \sin(377t) \).

Such derivatives help understand how quickly the magnetic field that surrounds an inductor change, determining how much and how quickly voltage will be induced. Mathematically, they represent the core of dynamic circuit analysis, translating real-world current fluctuations into impactful voltage changes.