Question

problem 21.21
a 420 - loop circular armature coil with a diameter of 6.0 cm rotates at 150 rev/s in a uniform magnetic field of strength 0.70 t
what is the rms voltage output of the generator?
express your answer to two significant figures and include the appropriate units.
part b
what would you do to the rotation frequency in order to double the rms voltage output?
to double the output voltage, you must decrease by two times the rotation frequency
to double the output voltage, you must triple the rotation frequency
to double the output voltage, you must increase by four times the rotation frequency
to double the output voltage, you must double the rotation frequency

Explanation:

Step1: Calculate the area of the coil

The formula for the area of a circle is \(A=\pi r^{2}\). Given the diameter \(d = 6.0\ cm=0.06\ m\), the radius \(r=\frac{d}{2}=0.03\ m\). So, \(A=\pi(0.03)^{2}= 2.827\times10^{-3}\ m^{2}\)

Step2: Find the angular frequency \(\omega\)

The rotation frequency \(f = 150\ rev/s\). The angular frequency \(\omega = 2\pi f\). So, \(\omega=2\pi\times150 = 942.48\ rad/s\)

Step3: Calculate the maximum emf \(\mathcal{E}_{0}\)

The formula for the emf of a rotating coil in a magnetic field is \(\mathcal{E}_{0}=NAB\omega\). Given \(N = 420\), \(B=0.70\ T\), \(A = 2.827\times10^{-3}\ m^{2}\), \(\omega=942.48\ rad/s\). Then \(\mathcal{E}_{0}=420\times2.827\times10^{-3}\times0.70\times942.48\approx790.7\ V\)

Step4: Calculate the rms voltage \(V_{rms}\)

The formula for rms voltage is \(V_{rms}=\frac{\mathcal{E}_{0}}{\sqrt{2}}\). So, \(V_{rms}=\frac{790.7}{\sqrt{2}}\approx560\ V\)

Step5: Analyze the relationship for part B

The emf formula is \(\mathcal{E}_{0}=NAB\omega\) and \(V_{rms}=\frac{\mathcal{E}_{0}}{\sqrt{2}}=\frac{NAB\omega}{\sqrt{2}}\). If \(V_{rms}\) is to be doubled (\(V_{rms,new} = 2V_{rms,old}\)), since \(N\), \(A\), \(B\) are constant, \(\omega_{new}=2\omega_{old}\). And since \(\omega = 2\pi f\), \(f_{new}=2f_{old}\)

Answer:

• \(V_{rms}=560\ V\)
• To double the output voltage, you must double the rotation frequency.