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Induced EMF — rotating loop

A rectangular loop of area A = 200 cm² rotates with angular velocity ω = 120π rad/s in a magnetic field B = 0.3 T.\nCalculate: (a) the peak EMF, (b) the RMS value of the EMF, (c) the rotation frequency.
Given data
A = 0.0200 m²ω = 120π ≈ 376.99 rad/sB = 0.3 T
Review the theory: Induzione EM
Steps
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  1. Calculate the peak EMF ε_max (in V).
  2. Calculate the RMS value of the EMF (in V).
  3. Calculate the rotation frequency f (in Hz).
Full worked solution
  1. Calculate the peak EMF ε_max (in V).
    εmax=NBAω=1⋅0.3⋅0.0200⋅120π\varepsilon_{max} = NBA\omega = 1 \cdot 0.3 \cdot 0.0200 \cdot 120\pi
    εmax⁡=NBAω=1×0.3×0.02×120π=0.3×0.02×120π=0.72π≈7.54 V\varepsilon_{\max} = N B A \omega = 1 \times 0.3 \times 0.02 \times 120\pi = 0.3 \times 0.02 \times 120\pi = 0.72\pi \approx \mathbf{7.54\,V}. The peak EMF is proportional to the magnetic field, area, and angular velocity.
  2. Calculate the RMS value of the EMF (in V).
    εrms=εmax2=7.542\varepsilon_{rms} = \dfrac{\varepsilon_{max}}{\sqrt{2}} = \dfrac{7.54}{\sqrt{2}}
    εrms=εmax⁡2=7.541.414≈5.33 V\varepsilon_{\mathrm{rms}} = \frac{\varepsilon_{\max}}{\sqrt{2}} = \frac{7.54}{1.414} \approx \mathbf{5.33\,V}. The RMS value is the effective value measured by an AC voltmeter.
  3. Calculate the rotation frequency f (in Hz).
    f=ω2π=120π2πf = \dfrac{\omega}{2\pi} = \dfrac{120\pi}{2\pi}
    f=ω2π=120π2π=60 Hzf = \frac{\omega}{2\pi} = \frac{120\pi}{2\pi} = \mathbf{60\,Hz}. The rotation frequency of 60 Hz matches the standard power grid frequency in many countries.
Result:ε_max = 7.54 V — ε_rms = 5.33 V — f = 60 Hz.
TheoryInduzione EMFormula sheetPhysics 2

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