# A simple generator is spinning with a frequency of 35 Hz, in a magnetic field of 0.15 T. It...

## Question:

A simple generator is spinning with a frequency of 35 Hz, in a magnetic field of 0.15 T. It consists of 22 loops with an area of 0.19 {eq}m^2 {/eq}, a length of 3.8 cm, and a resistance of 28 {eq}\Omega {/eq}.

(a) What is the equation for the voltage as a function of time?

(b) What is the rms voltage produced?

(c) What is the equation for the current as a function of time?

(d) What is the average power produced?

## EMF and Current Produced in a Generator

In generators EMF is produced by spinning a coil of N turns, having area A and of resistance R at some constant frequency F in a constant magnetic field B. In this case neither the area nor the magnetic field is time varying bu the angle between the plane of the coil and magnetic field varies with time. Due to this variation magnetic flux linked with the loop or coil changes and EMF and current will be induced. Magnetic flux linked with the coil is expressed as {eq}\phi = B A \cos \theta {/eq}. In this equation angle between the magnetic field and plane of the coil is related to time t as {eq}\theta = (2 \pi F) t = \omega t {/eq}.

Given points

• Spinning frequency of the generator F = 35 \ \ Hz
• Strength of the magnetic field B = 0.15 T
• Number of loops N = 22
• Area of the loops {eq}A = 0.19 \ \ m^2 {/eq}
• Total resistance {eq}R = 28 \ \ \Omega {/eq}

Part a)

EMF induced {eq}E = - N \dfrac { d \phi } { d t } \\ = - N B A \dfrac { d ( \cos \omega t ) } { d t } \\ = N B A \omega \sin ( \omega t ) {/eq}

Part b)

Maximum Induced EMF produced {eq}E = N B A \omega \\ = 22 \times 0.15 \times 0.19 \times ( 2 \pi \times 35 ) \\ = 137.885 \ \ V {/eq}

RMS voltage produced {eq}E_{rms} = \dfrac { E } { \sqrt { 2 } } \\ = \dfrac { 137.885 } { \sqrt { 2 } } \\ = 97.499 \ \ V {/eq}

Part c)

Induced current {eq}I = \dfrac { N B A \omega } { R } \times \sin ( \omega t ) {/eq}

Part d)

RMS current {eq}I_{rms} = \dfrac { N B A \omega } { \sqrt { 2 } R } \\ = \dfrac { 22 \times 0.15 \times 0.19 \times 2 \pi \times 35 } { \sqrt { 2 } \times 28 } \\ = 3.4821 \ \ A {/eq}

Average power produced {eq}P_{av} = I_{rms } \times V_{rms} \\ = 3.4821 \times 97.499 \\ = 339.501 \ \ W {/eq}