Question

Deduce an expression for the induced emf in the coil of the generator.

Hint:

For an a.c. generator, always remember: \[ \Phi = NBA\cos\omega t \] Applying Faraday's law, \[ e=-\frac{d\Phi}{dt} \] gives \[ e=E_0\sin\omega t \] with \[ E_0=NBA\omega \] This is one of the most important derivations from Electromagnetic Induction.

Answer 1

Concept: An alternating current generator works on the principle of electromagnetic induction. As the armature coil rotates in a uniform magnetic field, the magnetic flux linked with the coil changes continuously with time. According to Faraday's law, this change in magnetic flux induces an emf in the coil. The induced emf varies sinusoidally with time and therefore produces alternating current.

Step 1: Consider a rotating coil in a magnetic field Let
• \(N\) = number of turns in the coil,
• \(A\) = area of each turn,
• \(B\) = magnitude of the uniform magnetic field,
• \(\omega\) = angular velocity of rotation of the coil,
• \(t\) = time elapsed. Suppose the normal to the plane of the coil makes an angle \(\theta\) with the magnetic field at any instant. As the coil rotates with angular velocity \(\omega\), \[ \theta=\omega t \]

Step 2: Determine the magnetic flux linked with the coil Magnetic flux through one turn of the coil is given by \[ \phi = BA\cos\theta \] Substituting \[ \theta=\omega t \] we obtain \[ \phi = BA\cos\omega t \] Since the coil contains \(N\) turns, the total magnetic flux linked with the coil is \[ \Phi = NBA\cos\omega t \]

Step 3: Apply Faraday's law of electromagnetic induction According to Faraday's law, \[ e=-\frac{d\Phi}{dt} \] Substituting the expression for magnetic flux, \[ e=-\frac{d}{dt}(NBA\cos\omega t) \] Since \(N\), \(B\) and \(A\) are constants, \[ e=-NBA\frac{d}{dt}(\cos\omega t) \] Differentiating, \[ e=-NBA(-\omega\sin\omega t) \] \[ e=NBA\omega\sin\omega t \]

Step 4: Define maximum induced emf The quantity \[ NBA\omega \] is constant for a given generator. Let \[ E_0 = NBA\omega \] where \(E_0\) is called the maximum or peak value of the induced emf. Therefore, \[ e=E_0\sin\omega t \]

Step 5: Interpretation of the equation The equation \[ e=E_0\sin\omega t \] shows that:
• The induced emf varies sinusoidally with time.
• The emf changes its sign periodically.
• The direction of current reverses after every half cycle.
• The output of the generator is alternating in nature. When \[ \sin\omega t = 1 \] the emf becomes maximum: \[ e=E_0 \] When \[ \sin\omega t = 0 \] the induced emf becomes zero. Final Result: The instantaneous emf induced in the coil of an a.c. generator is \[ \boxed{e=E_0\sin\omega t} \] where \[ \boxed{E_0=NBA\omega} \] is the maximum value of induced emf. Thus, \[ \boxed{e=(NBA\omega)\sin\omega t} \] is the required expression for the induced emf in the rotating coil of an a.c. generator.