State the principle of an ac generator and explain its working with the help of a labelled diagram. Obtain the expression of the emf induced in a coil having N turn each of cross-sectional area A, rotating with the constant angular speed in a magnetic field ω, directed perpendicular to the axis of rotation.

Hint: In this question we have been asked to state the general principle of a generator and also explain its working. Therefore, we shall discuss the ac generators and with the help of a labelled diagram explain its working. A generator is a machine that converts mechanical energy into electrical energy.

Formula used:

[latex]E=−\frac{dϕ}{dt}[/latex]

Complete answer:

AC generators operate based on Faraday’s law of electromagnetic induction. This law indicates that an electromotive force is produced in a current-carrying loop situated within a uniform magnetic field. When a coil is rotated around its axis, which is perpendicular to the uniform magnetic field, the magnetic flux through the coil varies, resulting in an induced emf being generated across it ends.

When the coil in the generator starts moving, the arm AB starts moving upwards and the arm CD moves downwards. These arms begin to intersect the magnetic field lines; thus, according to Fleming’s right-hand rule, the induced current generated in these arms flows in the direction of AB and CD.

Let us consider a coil of N turns and area A that is rotated at constant angular velocity in a magnetic field of B as shown in the figure. When the normal to the coil is at an angle θ, magnetic flux is generated in the coil. This flux is given by,

ϕ=BANcosθ

We know that,

[latex]E=−\frac{dϕ}{dt}[/latex]

Substituting the value of Φ in above equation

We get,

⇒[latex]E=−\frac{dBANcos⁡θ}{dt}[/latex]

Now, we know that

θ=ωt

Therefore,

⇒[latex]E=−\frac{dBANcos⁡ωt}{dt}[/latex]

On solving,

⇒[latex]E=BANωsinωt[/latex]

It is understood that the peak value of the induced emf will be achieved when…

⇒[latex]θ=ωt=90^∘[/latex]

Therefore, the maximum value of induced emf will be E=BANω 

Note:

The device that produces electrical energy by transforming mechanical energy is referred to as an AC generator. The electrical energy generated takes the form of an alternating current represented as a sinusoidal waveform. The mechanical energy required for this conversion is typically provided by combustion engines, turbines, and similar sources.