22.2 FARADAY’S LAW
22.2.1 Faraday’s Law of Electromagnetic Induction
1 Inducing an emf in a plane coil: a demonstration
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| Figure 22.6 Shows a coil connected to a zero-centre galvanometer with no current flowing initially. |
Apparatus:
Refer to the apparatus shown in Figure 22.6. The series circuit consists of a coil of many turns and a zero-centre sensitive galvanometer. It does not contain a battery and so no current could be found in the circuit. The pointer of the galvanometer indicates a reading of zero.
Method:
Electromagnetic Induction: Procedure, Observation and Deduction
| Procedure | Observation | Deduction |
|---|---|---|
| Do not move the coil and the magnet. | The pointer does not deflect. | No emf is induced if there is no relative motion. |
| Move the magnet towards the coil. | Pointer deflects to the right. | Emf is induced due to relative motion. |
| Move the magnet away from the coil. | Pointer deflects to the left. | Emf is induced and polarity reverses. |
| Move the magnet faster. | Pointer deflects more. | Greater speed produces larger emf. |
| Move the coil towards the magnet. | Pointer deflects to the right. | Emf is induced due to relative motion of coil. |
| Move the coil away from the magnet. | Pointer deflects to the left. | Emf is induced and polarity reverses. |
Conclusion:
(a) When the coil and magnet are not moving, no emf exists in the circuit.
(b) Whenever the magnet or coil moves relative to the other, an emf is induced in the circuit.
(c) The magnet’s or coil’s
(i) direction of motion determines the direction of flow of the induced current
(ii) relative speed determines the magnitude of the induced current.
The higher the speed, the greater would be the current.
This implies that
(a) the magnitude of the induced emf is determined by the relative speed
(b) the polarity of the emf changes whenever there is a change in direction of relative motion.
2. Model used to explain observation
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| Figure 22.7 (a) No motion → no change in flux → no emf. (b) Magnet moves away → flux decreases → emf is induced. |
1. No relative motion:
Suppose the magnet is at rest in position X, and in this position we assume that there are 7 magnetic field lines linking the coil (this is just a model), as shown in Figure 22.7(a). Since the magnet is not moving, the flux linkage does not change with time. There is no current in the circuit.
Whenever there is no change of flux linkage through a coil, no emf is induced in the circuit.
2. Magnet moving:
(a) Next, the magnet moves away from the coil. At a particular instant the magnet is in position Y, as shown in Figure 22.7(b). Now, in this new position, the number of field lines linking the coil has decreased and is less than 7.
(b) Hence, as the magnet moves away from the coil, the flux linkage through the coil decreases with time. Now an emf is induced and an induced current flows in the coil.
(c) If the magnet moves away faster than before, the flux linkage decreases at a faster rate. The induced emf is larger, producing a larger induced current.
Whenever there is a change of flux linkage, an emf is induced in the circuit.
The greater the rate of change of flux linkage, the larger will be the induced emf.
We can perform an experiment using more elaborate apparatus to show that
rate of change of flux linkage ∝ magnitude of induced emf
This relationship is, in essence, Faraday’s law of electromagnetic induction.
3 Faraday’s law of electromagnetic induction
It states that:
The magnitude of the emf induced in a circuit is directly proportional to the rate of change of magnetic flux linkage through the circuit.
4 Equation from the law
According to the law, we have
magnitude of induced emf (E) ∝ rate of change of magnetic flux linkage [ d/dt (Φ) ]
E = −k d/dt (Φ)
where the negative sign is for taking into account Lenz’s law and k is a constant.
If we define E = 1 V for d/dt (Φ) = 1 Wb s−1, then k = 1. Furthermore,
Φ̅ = NΦ
where Φ is the magnetic flux through one turn of the coil. Hence, we have
E = −N dΦ/dt
EXAMPLE 22.6
The magnetic flux through a coil found inside the field varies with time in the manner as shown in Figure 22.8. Sketch a graph to show how the emf induced in the coil varies with time.
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| Figure 22.8 Shows how changing flux induces emf depending on its rate of change. |
Answer
Figure 22.9 shows how the induced emf in the coil varies with time. Notice the following:
(a) Time interval Δt1 < Δt2.
(b) The change of flux ΔΦ in both time intervals is the same.
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Figure 22.9 Shows that faster flux change (smaller Δt) produces larger induced emf. |
(c) Induced emf E = −N dΦ/dt. Hence, the flux must have changed at a faster rate during Δt1 than during Δt2.
(d) The maximum emf induced during Δt1 must be greater than the maximum induced during Δt2.
EXAMPLE 22.7
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| Figure 22.10 Graph showing how magnetic flux (Φ) changes with time, illustrating Faraday’s Law of Electromagnetic Induction where the rate of change of flux determines the induced emf. |
Answer
Figure 22.11 shows how the induced emf varies with time.
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Figure 22.11 Graph showing how the induced emf varies with time based on the rate of change of magnetic flux, illustrating Faraday’s Law of Electromagnetic Induction. |
EXAMPLE 22.8
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Figure 22.12 (a) A small coil placed inside a solenoid with its plane perpendicular to the magnetic field. |
A small coil is placed in the middle of a long solenoid with its plane perpendicular to the axis of the solenoid, as shown in Figure 22.12(a). The solenoid is connected to an electronic circuit which can produce alternating current whose waveform is as shown in Figure 22.12(b). Sketch a graph to show how the emf induced in the coil varies with time.
Answer
The magnetic flux density B at the centre of the solenoid is given by
B = μ0nI
Magnetic flux Φ is given by
Φ = BA cos θ
= (μ0nA cos θ)I
Φ ∝ I
dΦ/dt ∝ dI/dt
induced emf ∝ − dΦ/dt
∝ − dI/dt = −gradient of current-time curve
Figure 22.13 shows how the induced emf varies with time.
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| Figure 22.13 Graph showing how the induced emf varies with time due to a changing current in the solenoid, illustrating Faraday’s Law of Electromagnetic Induction where emf is proportional to the rate of change of current. |
EXAMPLE 22.9
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| Figure 22.14 A magnet falling through a horizontal coil produces a changing magnetic flux, inducing an emf that varies with time according to Faraday’s Law of Electromagnetic Induction. |
A magnet held vertically is released from rest. It passes through a horizontal coil, as shown in Figure 22.14. Sketch a graph to show how the emf induced in the coil varies with time as the magnet passes through the coil.
Answer
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| Figure 22.15 Graph showing how the induced emf varies as a magnet falls through a coil. The increasing speed of the magnet causes a faster change in magnetic flux, resulting in a larger emf, illustrating Faraday’s Law of Electromagnetic Induction. |
Figure 22.15 shows how the emf induced in the coil varies with time. The graph has the following features:
(a) As the magnet falls, its speed increases with time because it is pulled downwards by gravitational force. The speed after passing through the coil is faster than the speed before passing through the coil. Hence, t1 > t2.
(b) Since the speed is higher when the magnet is about to pass through the coil completely, the rate of change of flux linkage is higher. The emf induced within time interval $t_2$ is higher. Hence, $E_1 < E_2$.
(c) When the magnet approaches the coil, the flux linkage through the coil increases with time. But when the magnet has passed completely through the coil, the magnet moves away form the coil. Now the flux linkage decreases with time. Because of this, the polarity of the induced emf has to reverse when the magnet is half way through the coil.
(d) The area within time interval $t_1$ is equal to the area within time interval $t_2$.
EXAMPLE 22.10
A rectangular coil is placed inside a uniform magnetic field which points out of the page, as shown in Figure 22.16. Adjacent to this field is another uniform field which has the same strength but points into the page. The coil is made to move at constant speed from one field into the other in a direction which is perpendicular to the field. Sketch a graph to show how the induced emf in the coil varies with distance x travelled through the fields.
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| Figure 22.16 A rectangular coil moves at constant speed between two uniform magnetic fields in opposite directions, causing a change in magnetic flux and inducing an emf according to Faraday’s Law of Electromagnetic Induction. |
Answer
(a) As long as the coil is completely inside the field on the left, no emf is induced even though the coil is moving. The reason is that there is no change of magnetic flux linkage through the coil since the field is uniform.
(b) The moment one side of the coil starts to move into the other field, an emf is induced in the coil. The reason is that now there is a change of magnetic flux linkage through the coil since the second field points downwards while the first field points upwards.
(c) The induced emf is produced as long as the coil lies in both fields.
(d) When the coil is completely in the field on the right, the flux linkage through the coil does not change anymore. No emf is induced in the coil.
Figure 22.17 shows how the induced emf E varies with the distance x travelled through the field.
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| Figure 22.17 Graph showing how the induced emf varies with distance as the coil moves between two magnetic fields. Emf is induced only when the coil is in both fields, illustrating Faraday’s Law of Electromagnetic Induction. |
EXAMPLE 22.11
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| Figure 22.18 Graph showing how magnetic flux density (B) varies with time for a circular coil, illustrating Faraday’s Law of Electromagnetic Induction where changes in B induce an emf in the coil. |
A circular coil of diameter 2.0 cm has 10 turns. It is placed in a magnetic field whose flux density B varies with time in the manner as shown in Figure 22.18. The field is perpendicular to the plane of the coil. Sketch a graph to show how the emf E induced in the coil varies with time.
Answer
0 - 3.0 ms:
$\Phi =BA \ cos \ \theta$
$\frac{d\Phi}{dt}=A \ cos \ \theta \ \frac{dB}{dt}$
$=(\frac{1}{4}\pi d^2)(cos \ \theta)(gradient \ of \ line)$
$=(\frac{1}{4}\pi)(2.0 \times 10^{-2})^2\left(\frac{300-0}{3.0-0}\right)$
$=(3.14 \times 10^{-4})(1\times 10^2)$
$=3.14 \times 10^{-2} \ Wb \ s^{-1}$
$E_1=-N\frac{d\Phi}{dt}$
$=-(10)(3.14 \times 10^{-2})=-3.14 \times 10^{-1} \ V=-0.31 \ V$
3.0 - 4.0 ms:
$E_2=-(10)(3.14 \times 10^{-2})\left(\frac{0-300}{4.0-3.0}\right)=+9.4 \times 10^{-1} \ V=+0.94 \ V$
Figure 22.19 shows how the induced emf varies with time.
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| Figure 22.19 Graph showing how the induced emf varies with time due to changing magnetic flux, illustrating Faraday’s Law of Electromagnetic Induction. |
22.2.2 Average Induced emf
Flux Φ is given by Φ = BA cos θ
Hence, flux change can be brought about in the following ways:
(a) Changing B only ΔΦ = (Bf − Bi)A cos θ
(b) Changing A only ΔΦ = B(Af − Ai) cos θ
(c) Changing θ only ΔΦ = BA (cos θf − cos θi)
EXAMPLE 22.12
A rectangular coil with 10 turns is placed inside a magnetic field. Figure 22.20 shows how the magnetic flux through the coil varies with time. (a) Estimate the average emf induced in the coil. (b) Sketch a graph to show roughly how the induced emf varies with time.
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| Figure 22.20 Graph showing the variation of magnetic flux through a coil with time, used to estimate the induced emf and its trend based on Faraday’s Law of Electromagnetic Induction. |
Answer
(a) Initial flux Φi = +250 mWb
Final flux Φf = −250 mWb
Change of flux ΔΦ = Φf − Φi = (−250) − (+250) = −500 mWb
Time interval Δt = 4.0 − 3.0 = 1.0 ms
Average induced emf Eav = −N (ΔΦ / Δt)
= −(10) (−500 mWb / 1.0 ms) = +5 000 V
(b) Figure 22.21 shows how the induced emf varies with time.
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| Figure 22.21 Graph showing how the induced emf varies with time due to changes in magnetic flux, illustrating Faraday’s Law of Electromagnetic Induction. |
EXAMPLE 22.13
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| Figure 22.22 A rectangular coil being pulled out of a uniform magnetic field, causing a change in magnetic flux and inducing an emf according to Faraday’s Law of Electromagnetic Induction. |
A rectangular coil has 5 turns and size 2.0 cm × 3.0 cm. It is placed inside a uniform magnetic field with its plane perpendicular to the field, as shown in Figure 22.22. The field has flux density 300 mT. If the coil is pulled out of the field completely, producing an average induced emf of 0.20 V in the coil, determine the time taken to pull the coil completely out of the magnetic field.
Answer
Initial flux Φi = BAi cos θ
Final flux Φf = BAf cos θ
Change of flux ΔΦ = Φf − Φi = BAf cos θ − BAi cos θ = B(Af − Ai) cos θ = (0.300)[(0) − (2.0 × 3.0 × 10−4)] cos 0° = −1.8 × 10−4 Wb
Average induced emf Eav = −20 mV
Eav = −N (ΔΦ / Δt)
−0.20 = −(5) (1.8 × 10−4 / Δt)
Δt = 4.5 × 10−3 s
EXAMPLE 22.14
A rectangular coil of size 10 cm × 15 cm has 100 turns. It is placed inside a uniform magnetic field of flux density 20 mT. Initially, the plane of the coil is perpendicular to the field. If the coil is rotated through 90° in time interval 0.10 s, determine the average emf induced in the coil.
Answer
Initial flux through coil Φi = BA cos θi
Final flux through coil Φf = BA cos θf
Flux change ΔΦ = Φf − Φi = BA(cos θf − cos θi) = (20 × 10−3)(0.10 × 0.15)(cos 90° − cos 0°) = −3.0 × 10−4 Wb
Eav = −N (ΔΦ / Δt)
= −100 (3.0 × 10−4 / 0.10) = +0.30 V
EXAMPLE 22.15
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| Figure 22.23 Two concentric circular coils in the same plane, where a changing current in the larger coil produces a changing magnetic flux that induces an emf in the smaller coil, illustrating Faraday’s Law of Electromagnetic Induction. |
Two concentric circular coils lie on the same plane, as shown in Figure 22.23. The larger coil has radius r1 = 5.0 cm and resistance R1 = 2.0 Ω. The smaller coil has radius r2 = 2.0 cm. Each coil has a single turn. A battery of emf E = 12 V and internal resistance R = 1.0 Ω is connected to the larger coil. Estimate the average emf induced in the smaller coil if the battery is disconnected and the current is reduced to zero in 2.0 ms.
Answer
The current I flowing in the large coil is given by E = I(R1 + R)
I = 12 / (2.0 + 1.0) = 4.0 A
This current produces a magnetic field of flux density Bi. The field passes perpendicularly through the small coil. We make the assumption that the magnetic field on the plane of the coil is uniform, and the flux density is given by
Bi = μ0I / 2r1
= (4π × 10−7)(4.0) / 2(5.0 × 10−2)
= 5.03 × 10−5 T
When the current becomes zero, the flux density becomes
Bf = 0
Change of magnetic flux through the smaller coil is
ΔΦ = Φf − Φi
= BfA cos θ − BiA cos θ
= (Bf − Bi)A cos θ
= (0 − 5.03 × 10−5)(π × (2.0 × 10−2)2) cos 0°
= −6.32 × 10−8 Wb
Average induced emf
Eav = −N (ΔΦ / Δt)
= −(1) (−6.32 × 10−8 / 2.0 × 10−3)
= +3.2 × 10−5 V
| No | Application | Description | Key Concept |
|---|---|---|---|
| 1 | Electric Generators | Convert mechanical energy into electrical energy in power plants. | Rotation changes magnetic flux → induces emf |
| 2 | Transformers | Increase or decrease voltage in power transmission systems. | Changing flux between coils |
| 3 | Induction Cookers | Heat cookware directly without flame. | Changing current → magnetic field → heat |
| 4 | Electric Guitar Pickups | Convert string vibrations into electrical signals. | Flux change from vibrating strings |
| 5 | Wireless Charging | Transfers energy without wires. | Coil-to-coil electromagnetic induction |
| 6 | Induction Motors | Used in appliances and industry. | Rotating magnetic field induces current |
| 7 | Metal Detectors | Detect hidden metallic objects. | Induced currents in metals |
| 8 | Magnetic Braking | Slows motion without contact. | Opposing magnetic fields from induced current |
| 9 | Dynamos | Generate electricity from motion (e.g., bicycles). | Moving magnet induces current |
| 10 | MRI Machines | Medical imaging technology. | Strong magnetic fields + induced signals |
Faraday’s Law of Electromagnetic Induction explains how a changing magnetic flux can produce an induced electromotive force (emf). This fundamental principle is essential in understanding how electrical energy is generated and utilized in modern technology.
From electric generators and transformers to wireless charging and medical imaging systems, the applications of this law are widely used in everyday life. These technologies rely on the ability to control and manipulate changes in magnetic flux efficiently.
In summary, the greater the rate of change of magnetic flux linkage, the larger the induced emf produced. This makes Faraday’s Law a key concept in both physics and engineering, especially in the development of energy systems and electronic devices.
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