To monitor the breathing of a hospital patient, a thin belt...

To monitor the breathing of a hospital patient, a thin belt is wrapped around the patient's chest. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by $39.0 \mathrm{~cm}^{2}$. The magnitude of the Earth's magnetic field is $50.0 \mu \mathrm{T}$ and makes an angle of $28.0^{\circ}$ with the plane of the coil. If a patient takes $1.80 \mathrm{~s}$ to inhale, find the average induced emf in the coil during this time.

Key Concepts

Faraday's Law of Electromagnetic Induction

This law states that a time-varying magnetic flux through a closed loop induces an electromotive force (emf) in the loop. It is the foundational principle behind devices such as transformers, generators, and inductive sensors, and mathematically, it is expressed as the induced emf being equal to the negative rate of change of magnetic flux through the circuit.

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing through a given area. It is calculated by taking the product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the magnetic field direction and the normal (perpendicular) to the surface. It captures how much magnetic field is 'captured' by the surface.

Time Rate of Change of Flux

This concept involves determining how fast the magnetic flux through a loop changes over time. In electromagnetic induction, the rate of change of the flux is key because it directly influences the magnitude of the induced emf, making it critical for understanding dynamic changes in circuits exposed to varying magnetic environments.

Multiple-Turn Coil Configuration

In a coil with several turns, the total induced emf is the sum of the emfs induced in each individual turn. This means that the induced emf is multiplied by the number of turns, enhancing the overall effect of any change in magnetic flux. This concept is crucial in the design and functioning of devices like transformers and sensors.

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