A big circular coil with $1000$ turns and an average radius of $10 ext{ m}$ is rotating about its horizontal

Question

A big circular coil with $1000$ turns and an average radius of $10 	ext{ m}$ is rotating about its horizontal diameter at a rate of $2 	ext{ rad s}^{-1}$. The vertical component of the Earth's magnetic field at that location is $2 	imes 10^{-5} 	ext{ T}$, and the electrical resistance of the coil is $12.56 \Omega$. The maximum induced current in the coil will be:

Accepted Answer

The maximum induced electromotive force ($\varepsilon_{max}$) in a rotating coil is given by the formula $\varepsilon_{max} = NBA\omega$, where:
*   $N$ is the number of turns ($1000$).
*   $B$ is the magnetic field ($2 	imes 10^{-5} 	ext{ T}$).
*   $A$ is the area of the coil ($\pi r^2 = \pi (10)^2 = 100\pi 	ext{ m}^2$).
*   $\omega$ is the angular velocity ($2 	ext{ rad s}^{-1}$).

The maximum induced current ($I_{max}$) is given by Ohm's law: $I_{max} = \frac{\varepsilon_{max}}{R}$.

Substituting the values:
$I_{max} = \frac{1000 	imes (2 	imes 10^{-5}) 	imes (100\pi) 	imes 2}{12.56}$

Noting that $12.56 \approx 4\pi$:
$I_{max} = \frac{1000 	imes 2 	imes 10^{-5} 	imes 100\pi 	imes 2}{4\pi}$
$I_{max} = \frac{400000 \pi 	imes 10^{-5}}{4\pi}$
$I_{max} = \frac{4\pi}{4\pi} = 1 	ext{ A}$.

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