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Electrical Generators and Motors

A generator has a cell with 1000 circular loops of wire that each have a radius of 30 cm. They have a frequency of 60 Hz. The generator will create an rms (average) voltage of 100 V with a maximum electromotive force (emf) of 150 V. What is the magnitude of the magnetic field?

Given:

- $\varepsilon_{0}=150\ V$
- $r=0.30\ m$
- $N=1000$
- $f=60\ Hz=60\ s^{-1}$

Find:

- B (magnetic field)

$\varepsilon =N\cdot B\cdot A\cdot\omega\cdot sin(\omega t)$

To maximize this equation, we must look at the sin(ωt) part.

Sin(x) can be at most equal to one.

$\varepsilon_{0}=N\cdot B\cdot A\cdot \omega\cdot (1)$

- Angular frequency: $\omega =2\pi f$
- Area: $A=\pi r^2$

$\frac{\varepsilon_{0}}{N(\pi r^{2})(2\pi f)}=\frac{N\cdot B\cdot (\pi r^{2})(2\pi f)}{N(\pi r^{2})(2\pi f)}$

$B=\frac{\varepsilon_{0}}{2N(\pi^{2}r^{2}f)}$

$B=\frac{150\ V}{2\pi^{2}(1000)(0.30\ m)^{2}(60\ s^{-1})}=(1.41\times 10^{-3}\frac{V\cdot s}{m^2})(\frac{1\ T}{1\ \frac{V\cdot s}{m^{2}}})$

$B=1.41\times 10^{-3}\ T$