Two coils are close to each other. The first coil carries a...

Two coils are close to each other. The first coil carries a current given by $I(t)=5.00 e^{-0.0250 t} \sin 120 \pi t,$ where $I$ is in amperes and $t$ is in seconds. At $t=0.800 \mathrm{s}$ , the emf measured across the second coil is $-3.20 \mathrm{V}$ . What is the mutual inductance of the coils?

Key Concepts

Mutual Inductance

Mutual inductance is a measure of the ability of one current-carrying coil to induce an electromotive force (emf) in another nearby coil when the current in the first coil changes. It quantifies the coupling between the two coils' magnetic fields and is fundamental to applications like transformers and inductive sensors. The mutual inductance (M) is defined by the relation emf = -M (dI/dt), where dI/dt is the rate of change of current in the primary coil.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a changing magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This principle provides the foundation for understanding how time-varying currents in one coil induce voltage in a second coil. In the context of mutual inductance, the induced emf in a coil due to a changing current in a neighboring coil is directly proportional to the rate of change of that current and the mutual inductance between the coils.

Differentiation of Time-Varying Functions

When dealing with time-varying currents, especially those that involve functions like exponentials and sinusoidal terms, it is necessary to compute the derivative of the current with respect to time. This derivative represents the rate of change of the current, which is used in Faraday's law to determine the induced emf. The process often involves applying rules from calculus, including the product rule and chain rule, to accurately determine the rate at which the current changes over time.

Transcript

Please give Ace some feedback