AC Generator

Answers & Solutions

Work the questions above first, then check yourself here. Each answer is set out the way it should be written in the paper.

1. 1.↑ back to the question

   Principle: An ac generator (dynamo) is a device that converts mechanical energy into electrical energy. It works on the principle of electromagnetic induction — whenever the magnetic flux linked with a coil changes, an emf is induced in it. Working: The armature coil (of $N$ turns, area $A$) is rotated with a constant angular speed $\omega$ inside a uniform magnetic field $B$, kept fixed. As the coil rotates, the angle between the area vector and $B$ keeps changing, so the flux linked with the coil changes, inducing an alternating emf, which is picked off via slip rings and brushes connected to the external circuit. If $\theta = \omega t$ is the angle between $\vec{B}$ and the area vector of the coil at time $t$, the flux linked is $\phi = NBA\cos\omega t$ By Faraday's law, induced emf, $\varepsilon = -\dfrac{d\phi}{dt} = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t$, where $\varepsilon_0 = NBA\omega$ is the peak (maximum) value of the induced emf.

2. 2.↑ back to the question

   (a) Principle: an ac generator works on the principle of electromagnetic induction — when a coil is rotated in a uniform magnetic field, an emf is induced in it. If the coil (of $N$ turns, area $A$) rotates with angular speed $\omega$ in field $B$, the flux linked at time $t$ is $\phi = NBA\cos\omega t$ Induced emf, $\varepsilon = -\dfrac{d\phi}{dt} = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t = \varepsilon_0\sin 2\pi f t$, where $\varepsilon_0 = NBA\omega = 2\pi f NBA$, and $f$ is the frequency of rotation of the coil. (b) Slip rings are used to provide a movable (sliding) contact between the rotating armature coil and the external circuit containing the load.

3. 3.↑ back to the question

   (a) The ac generator consists of an armature coil rotated by some external mechanical means in a uniform magnetic field between the poles of a strong field magnet. The rotating armature is connected to the external circuit via slip rings and brushes. As the coil rotates, the flux linked with it changes as $\phi = BA\cos\omega t$, so the induced emf is $\varepsilon = -\dfrac{d\phi}{dt} = BA\omega\sin\omega t$; for a coil of $N$ turns, $\varepsilon = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t$. This varies sinusoidally with time, so the emf generated is alternating. (b) Given $r = 10\text{ cm} = 0.1\text{ m}$, so $A = \pi r^2 = 3.14\times (0.1)^2 = 0.0314\text{ m}^2$; $N = 20$; $\omega = 50\text{ rad s}^{-1}$; $B = 3.0\times 10^{-2}\text{ T}$ (i) Maximum induced emf, $\varepsilon_{max} = NAB\omega = 20\times 0.0314\times 3.0\times 10^{-2}\times 50 = 0.942\text{ V}$ Average induced emf (over the range of the cycle, taken as the mean of the maximum and minimum values), $\varepsilon_{avg} = \dfrac{\varepsilon_{max} + \varepsilon_{min}}{2} = \dfrac{0.942 + 0}{2} = 0.471\text{ V}$ (ii) Given $R = 10\,\Omega$. Maximum current in the coil, $I_0 = \dfrac{\varepsilon_0}{R} = \dfrac{0.942}{10} = 0.0942\text{ A}$ Average current, $I_{avg} = \dfrac{\varepsilon_{avg}}{R} = \dfrac{0.471}{10} = 0.0471\text{ A}$ Average power loss due to Joule heating, $P = I_{avg}^2 R = (0.0471)^2\times 10 = 0.022\text{ W}$

4. 4.↑ back to the question

   Principle: An ac generator works on the principle of electromagnetic induction — it converts mechanical energy into electrical energy. Construction: The ac generator consists of (i) an armature coil of a large number of turns of insulated copper wire, wound over a soft iron core (the soft iron core increases the magnetic flux linked with the coil), (ii) strong field magnets that provide the magnetic field, with the armature coil rotated between their pole faces (axis of rotation perpendicular to the field lines), (iii) slip rings, which provide a movable contact for the armature coil with the external circuit, and (iv) carbon brushes, which pass the current from the armature coil on to the external circuit containing the load. Working: As the armature coil is rotated in the magnetic field, the orientation of the coil relative to the field, and hence the magnetic flux linked with it, keeps changing. This changing flux induces an emf in the coil (Faraday's law). If $\theta = \omega t$ is the angle turned through at time $t$, the flux linked with the coil is $\phi = NBA\cos\omega t$ Induced emf, $\varepsilon = -\dfrac{d\phi}{dt} = NBA\omega\sin\omega t = \varepsilon_0\sin\omega t$, where $\varepsilon_0 = NBA\omega$ The direction of the induced current is given by Fleming's right-hand rule.

5. 5.↑ back to the question

   (a) [As derived above:] With the armature coil (area $A$, $N$ turns) rotating at angular speed $\omega$ in field $B$, the flux linked is $\phi = NBA\cos\omega t$, so the instantaneous induced emf is $\varepsilon = -\dfrac{d\phi}{dt} = NBA\omega\sin\omega t$ (b) Here, $A = 200\text{ cm}^2 = 2\times 10^{-2}\text{ m}^2$, $N = 20$, $\omega = 50\text{ rad s}^{-1}$, $B = 3\times 10^{-2}\text{ T}$ Maximum emf induced in the coil, $\varepsilon_0 = NBA\omega = 20\times 3\times 10^{-2}\times 2\times 10^{-2}\times 50 = 0.6\text{ V}$ If $R$ is the resistance of the coil, the maximum value of the current is $I_{max} = \dfrac{\varepsilon_0}{R} = \dfrac{0.6\text{ V}}{R}$

6. 6.↑ back to the question

   As derived above, when the coil rotates with angular speed $\omega$ in the field $B$, the induced emf is $\varepsilon = NBA\omega\sin\omega t$, so its maximum (peak) value is $\varepsilon_0 = NBA\omega$ Since the coil makes $f$ revolutions per second, its angular speed is $\omega = 2\pi f$ $\therefore \varepsilon_0 = NBA(2\pi f) = 2\pi f NBA$, as required.

7. 7.↑ back to the question

   (a) An ac generator consists of an armature coil of many turns of insulated copper wire wound on a soft iron core, rotated between the poles of strong field magnets, with slip rings providing a moving contact and carbon brushes carrying the current to the external circuit. Principle: it works on electromagnetic induction — rotating the armature coil in the magnetic field changes the flux linked with it, inducing an emf. Taking the plane of the coil to be initially perpendicular to the field (so flux is maximum at $t=0$), the flux varies as $\phi = BA\cos\omega t$, a cosine curve oscillating between $+BA$ and $-BA$ with peaks/zeros at $\omega t = 0, \pi, 2\pi,\ldots$ The induced emf, $\varepsilon = BA\omega\sin\omega t$, is a sine curve (zero at $t=0$, rising to its peak $\varepsilon_0 = BA\omega$ at $\omega t = \pi/2$, back to zero at $\omega t = \pi$, and so on), i.e. it lags the flux curve by a quarter cycle. (b) A choke coil (a pure inductor, ideally with negligible resistance) reduces the current in an ac circuit without dissipating any power, since the current and voltage across a pure inductor are out of phase by $90°$, making the average power consumed over a cycle zero. A rheostat can also reduce the current, but it does so by dissipating energy as heat, wasting power. Hence a choke coil is the economical choice for controlling the current to fluorescent tubes on ac mains.

8. 8.↑ back to the question

   (a) The ac generator has an armature coil (many turns of copper wire on a soft iron core), field magnets, slip rings and carbon brushes. Principle: electromagnetic induction — mechanical energy is converted into electrical energy by rotating the coil in a magnetic field. (b) In a generator, the armature coil is mechanically rotated while the field magnets are kept fixed. As the coil rotates, the orientation of its plane relative to the (fixed) field direction keeps changing, so the magnetic flux linked with it changes continuously. (c) If the coil ($N$ turns, area $A$) rotates with angular speed $\omega$, the flux is $\phi = NBA\cos\omega t$, so the induced emf is $\varepsilon = -\dfrac{d\phi}{dt} = NBA\omega\sin\omega t$ The maximum value of the induced emf is $\varepsilon_0 = NBA\omega$. The direction of the induced current is given by Fleming's right-hand rule. (d) Taking the coil's plane to be initially perpendicular to the field (i.e. flux maximum, emf zero at $t=0$), the emf $\varepsilon = \varepsilon_0\sin\omega t$ traces a sine curve against time: zero at $t=0$, rising to peak $\varepsilon_0$ at a quarter period, back to zero at half a period, down to $-\varepsilon_0$ at three-quarters of a period, and back to zero after a full period $T = 2\pi/\omega$.