22.2.5 Disc Rotating in Magnetic Field
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| Figure 22.40 (a) Cakram logam berputar dalam medan magnet B dan dihubungkan ke rangkaian luar melalui titik X dan Y. (b) Elemen jari-jari cakram menyapu medan magnet saat berputar, sehingga timbul emf antara pusat dan tepi cakram. |
1. The apparatus
Consider a circular metal disc of radius R which can rotate about a perpendicular axis. The axis is a conductor which passes through the centre O of the disc, as shown in Figure 22.40(a). The rim of the disc makes electrical contact with a conductor P. An external circuit can be connected to the disc via points X and Y.
A magnetic field is applied perpendicularly to a part of the disc. When the disc rotates, an emf will be induced in the disc. The induced emf is developed across O and P, and will appear across XY.
2. Equation for the induced emf
Imagine that the disc is composed of an infinite number of very fine wires. Each imaginary wire has a thickness which is the same as that of the disc, and lies on the radius of the disc.
As the disc rotates at constant angular velocity ω, one of those imaginary wires may happen to be sweeping across the uniform magnetic field, which is perpendicular to the wire and has strength B, as shown in Figure 22.40(b).
The wire undergoes angular displacement Δθ in time Δt, sweeping through area ΔA of the magnetic field. The magnetic flux through this area is
ΔΦ = B(ΔA) cos 0°
An emf E is induced across the ends of the wire, given by
E = −N (ΔΦ / Δt) = −(1)B (ΔA / Δt)
But,
ΔA / Δt = surface area of disc / period of revolution
period of revolution = 2π / ω
Hence,
ΔA / Δt = πR² / (2π / ω)
Induced emf
E = −B (½ R²ω)
= − ½ BR²ω
3. Application of the induced emf
Notice that
E ∝ ω
But,
ω = 2πf
where f is the frequency of revolution of the disc. Hence
E ∝ f
The apparatus as shown in Figure 22.40(a) can be attached to any revolving object, like a car wheel, motorcycle wheel, a shaft. The object and the disc both have a common axis of rotation. When the object rotates, hence rotating the disc, an induced emf will be produced by the rotating disc. The magnitude of the induced emf is proportional to the angular velocity of the object. The emf can be recorded by a voltmeter. The scale of the voltmeter can be adapted to read angular speed instead of voltage. Hence, this setup can become a simple form of a speedometer or tachometer.
EXAMPLE 22.21
Refer to the disc shown in Figure 22.40(a). It has a radius of 15.0 cm and is rotating at a constant angular velocity of 300 rpm. A uniform magnetic field of strength 300 mT passes perpendicularly through part of the surface of the disc. Determine the emf induced across the centre of the disc and the rim.
Answer
Angular velocity
ω = (300 × 2π rad) / 60 s = 31.4 rad s−1
Magnitude of induced emf
E = ½ BR²ω = ½ (0.300)(0.15)²(31.4) = 0.106 V
22.2.6 Plane Coil Rotating in Magnetic Field
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| Figure 22.41 (a) Kumparan ditempatkan di antara kutub magnet dan diputar sehingga memotong medan magnet. (b) Tampak samping menunjukkan sudut θ antara normal kumparan dan arah medan magnet saat kumparan berputar. |
1. Equation for induced emf
Consider a plane coil placed in a uniform magnetic field, as shown in Figure 22.41(a). When the coil rotates about an axis perpendicular to the field, an emf will be induced in the coil.
Suppose that the coil of N turns and surface area A rotates at constant angular velocity ω in the field of strength B. The side view of the position of the coil at a particular instant is shown in Figure 22.41(b). At that instant, the magnetic flux linkage Φ through the coil is
Φ = NΦ = N(BA cos θ)
where θ is the angle between the normal to the coil and the direction of the magnetic field. Since the angular velocity ω is constant, we have
θ = ωt
The induced emf E is given by
E = − dΦ/dt = − d/dt (NBA cos ωt)
But N, B and A are constants. Hence
E = −BAN d/dt (cos ωt)
The magnitude of the emf induced in the rotating coil is given by
E = BAN ω sin ωt
2. Maximum induced emf
The induced emf varies sinusoidally with time. The maximum voltage E0 obtainable is
E0 = BAN ω
because the maximum value of sin ωt is ±1. We may rewrite the equation for E as
E = E0 sin ωt
3. Graph of E against t
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| Figure 22.42 The graph shows how the induced emf E varies with time t. The emf is maximum when the coil cuts the magnetic field most effectively, and zero when the coil is parallel to the field. |
Figure 22.42 shows how the induced emf E varies with time t. The graph also shows the positions of the coil at instants when the emf is either minimum (zero) or maximum.
Take note of the following:
(a) The emf induced in a coil rotating in a uniform magnetic field varies sinusoidally with time.
(b) The emf is an alternating voltage because the voltage has positive values as well as negative values.
(c) Maximum voltage is produced at the instant when the coil is parallel to the magnetic field. No voltage exists at the instant when the coil is perpendicular to the magnetic field.
EXAMPLE 22.22
A rectangular coil of 200 turns has size 10 cm × 15 cm. It rotates at a constant angular velocity of 600 rpm in a uniform magnetic field of strength 200 mT. Determine
(a) the maximum voltage produced by the coil
(b) the voltage produced at the instant when the plane of the coil makes an angle of 60° with the field.
Answer
(a) Angular velocity
ω = (600 × 2π rad) / 60 s = 62.8 rad s−1
Maximum induced emf
E0 = BAN ω = (0.200)(0.10 × 0.15)(200)(62.8) = 37.7 V
(b) Instantaneous emf
E = BAN ω sin ωt = E0 sin θ
Given
θ = 90° − 60° = 30°
Hence,
E = 37.7 sin 30° = 18.9 V
EXERCISE 22.5
1 A square coil of length 10 cm and having 100 turns rotates at a constant angular velocity of 150 rpm in a uniform magnetic field of strength 5.0 mT. Determine the maximum emf induced in the coil.
| Applications of Induced EMF (Rotating Disc & Plane Coil) | ||
|---|---|---|
| Application | System Type | Description |
| Electric Generator | Plane Coil | Converts mechanical energy into electrical energy using rotating coils in a magnetic field. Widely used in power stations. |
| Alternator | Plane Coil | Produces alternating current (AC) by rotating a coil inside a magnetic field. Used in vehicles and electricity supply systems. |
| Speedometer / Tachometer | Rotating Disc | Measures rotational speed by generating emf proportional to angular velocity of a rotating disc. |
| Magnetic Flow Meter | Rotating Disc Concept | Measures flow rate of conductive fluids based on induced emf when fluid moves through a magnetic field. |
| Bicycle Dynamo | Plane Coil | Generates electricity for lights when the wheel rotates, applying electromagnetic induction. |
| Wind Turbine Generator | Plane Coil | Converts wind energy into electrical energy using rotating coils driven by turbine blades. |
| Hydroelectric Generator | Plane Coil | Uses flowing water to rotate turbines and generate electricity via electromagnetic induction. |
In a rotating disc, the induced emf depends on the angular velocity, while in a plane coil, the emf varies sinusoidally with time following the relationship E = BANω sin(ωt). This leads to the production of alternating current (AC).
These principles are widely applied in devices such as generators, alternators, and speedometers, where energy conversion is essential. Overall, electromagnetic induction ensures that mechanical energy is efficiently converted into electrical energy, supporting many real-world applications.
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