Physics equations/Faraday law/Faraday law example

Spinning coil in a magnetic field

Faraday's law of induction|Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.

$ℰ=-\frac{d{\mathrm{\Phi }}_B}{dt},$

where $ℰ$ is the electromotive force (emf) in volts and Φ_B is the magnetic flux in Weber (Wb)|webers. For a loop of constant area, A, spinning at an angular velocity of $\omega$ in a uniform magnetic field, B, the magnetic flux is given by

${\mathrm{\Phi }}_B=B\cdot A\cdot \cos (\theta ),$

where θ is the angle between the normal to the current loop and the magnetic field direction. Since the loop is spinning at a constant rate, ω, the angle is increasing linearly in time, θ=ωt, and the magnetic flux can be written as

${\mathrm{\Phi }}_B=B\cdot A\cdot \cos (\omega t).$

Taking the negative derivative of the flux with respect to time yields the electromotive force.

$ℰ=-\frac{d}{dt}[B\cdot A\cdot \cos (\omega t)]$	Electromotive force in terms of derivative
$=-B\cdot A\frac{d}{dt}\cos (\omega t)$	Bring constants (A and B) outside of derivative
$=-B\cdot A\cdot (-\sin (\omega t))\frac{d}{dt}(\omega t)$	Apply chain rule and differentiate outside function (cosine)
$=B\cdot A\cdot \sin (\omega t)\frac{d}{dt}(\omega t)$	Cancel out two negative signs
$=B\cdot A\cdot \sin (\omega t)\omega$	Evaluate remaining derivative
$=\omega \cdot B\cdot A\sin (\omega t).$	Simplify.