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Mutual inductance is a fundamental concept in electromagnetism that describes how the change in current in one coil can induce an electromotive force (emf) in another nearby coil. It is denoted by the symbol \( M \) and measured in henries (H). In our given exercise, the mutual inductance between two coils is \( M = 3.25 \times 10^{4} \) H. This indicates a strong coupling between the coils.

The concept of mutual inductance is crucial in understanding how transformers, induction coils, and many electrical devices operate. It allows for the transmission of electric energy between circuits without a physical connection.

• When the current in the first coil changes, the magnetic field created by this current also changes.
• This changing magnetic field then induces an emf in the second coil according to the same mutual inductance \( M \).
• Thus, mutual inductance is a measure of how efficiently these coils are influencing each other.

Faraday's Law of electromagnetic induction is the principle that quantifies the induced emf in a coil due to the change in magnetic flux. According to Faraday's Law, the induced emf in a coil is proportional to the rate of change of current in a neighboring coil. The expression is given by:

\[\text{Induced emf} = -M \cdot \frac{di}{dt}\]
Here, the negative sign indicates the direction of the induced emf, following Lenz's law, which tries to oppose the change in current. In our example, we calculate the induced emf using the given mutual inductance \( M = 3.25 \times 10^{4} \) H and the rate of change of current \( \frac{di}{dt} = 830 \mathrm{~A/s} \).

These calculations are straightforward:

• Calculate the product of mutual inductance \( M \) and the rate of change of current.
• Apply the formula to find the magnitude of the induced emf.
• Notably, if the current changes uniformly, the rate \( \frac{di}{dt} \) is constant, resulting in a constant induced emf.

A uniform rate of current increase means that the rate at which the current changes is consistent over time. In our exercise, the current in one of the coils increases uniformly at a rate of \( 830 \mathrm{~A/s} \). This consistency in the rate is crucial because:

• It ensures that the induced emf remains constant over time, as obtained from Faraday's Law.
• When an emf is induced due to a uniform change, it implies that the magnetic flux is changing at a constant rate, leading to a steady induced emf.
• This helps in simplifying the analysis and calculations related to electromagnetic induction, making these scenarios easier to handle in practical applications.

In practical situations, maintaining a uniform rate of change can be challenging, but it simplifies many real-world applications and analyses, such as in designing circuits and coils. These properties are leveraged in devices like transformers where a consistent change is crucial for reliable performance.