Astronauts on a space station decide to use Earth's magnetic...

Astronauts on a space station decide to use Earth's magnetic field to generate electric current. Earth's $\vec{B}$ field in this region has the magnitude of $3.0 \times 10^{-7}$ T. They have a coil that rotates $90^{\circ}$ in 1.2 s. The area inside the coil measures $5000 \mathrm{m}^{2}$. Estimate the number of loops needed in the coil so that during that $90^{\circ}$ turn it produces an average induced emf of about $120 \mathrm{V}$. Indicate any assumptions you made. Is this a feasible way to produce electric energy?

Key Concepts

Faraday's Law of Induction

This principle states that a change in magnetic flux through a coil will induce an electromotive force (emf) in that coil. The law quantitatively relates the emf to the rate of change of flux, meaning that faster or larger changes in flux will yield a higher induced voltage. It is a cornerstone concept in understanding how generators and transformers operate.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, adjusted by the cosine of the angle between the field and the normal to the area. It quantifies the total magnetic field 'captured' by a surface and is crucial for calculating induced emf when the orientation of the area changes with respect to a magnetic field.

Induced EMF and Coil Turns

The induced emf in a coil is directly proportional to both the rate of change of the magnetic flux and the number of turns in the coil. Multiple loops in a coil effectively multiply the total induced voltage by the number of turns, which is a key design parameter when trying to reach a target voltage in applications like generators.

Assumptions in Magnetic Induction Applications

Practical problems often assume a uniform magnetic field and neglect resistive losses or other inefficiencies, allowing for simplified calculations. In assessing the feasibility of energy generation, it is also important to consider that using weak ambient magnetic fields may require impractically large areas or numbers of loops, which influences the overall design and viability of the technology.

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