21.7 TORQUE ON A COIL
21.7.1 How Torque on a Current-carrying Coil is Produced
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| Figure 21.37 A rectangular coil ABCD is placed in a uniform magnetic field B, where sides AD and BC are perpendicular to the field, and the normal to the coil makes an angle θ with the magnetic field direction. |
Consider a rectangular coil ABCD which has one turn and has length AB = a and breadth BC = b. It is placed inside a uniform magnetic field of strength B, as shown in Figure 21.37. The side AD (and also BC) of the coil is perpendicular to the direction of the field. In general, the normal to the plane of the coil may make an angle θ with the direction of the field.
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| Figure 21.38 A plan view of coil ABCD showing current flowing into the page from B to C and out of the page from D to A when a voltage is applied. |
Suppose a voltage is applied to the coil, causing a current I to flow in the coil in the direction ABCDA. Figure 21.38 shows a plan view of the coil at the moment when the voltage is applied. It shows current flowing into the page (downwards from B to C) in one part of the coil and also current flowing out of the page (upwards from D to A).
At the moment when current starts to flow in the coil, each length (AB, BC, CD, DA) of the coil is acted on by a magnetic force. Consider the following lengths:
(a) AD and BC:
(i) Since AD and BC both carry current and are inside a magnetic field, they are acted on by magnetic forces.
(ii) In order to determine the directions of the magnetic forces, we apply Fleming’s left hand rule to AD and also to BC. Figure 21.35(b) shows the direction of the force (F) that acts on AD and also the direction of the force (−F) that acts on BC. The two forces are perpendicular to the magnetic field.
(iii) The two magnetic forces are parallel to each other, opposite in direction and separated from each other. This means that the pair of forces forms a couple. Hence, at the moment when current starts to flow in the coil, a couple is produced. The couple in turn produces a torque which acts on the coil and tends to rotate it about an axis. The axis passes through the centre of the coil, lies on the plane of the coil and is parallel to AD and BC. Referring to the plan view shown in Figure 21.35(b), we notice that the couple tends to rotate the coil in the anti-clockwise direction.
(b) AB and CD:
Each length carries current and is inside a magnetic field. Hence, each is also acted on by magnetic force. We can use Fleming’s left hand rule to show that both forces lie on the plane of the coil, and they point towards each other. Since they lie on the same plane, the two forces do not produce a couple as long as AB and CD always remain perpendicular to the field.
21.7.2 Magnitude of the Torque in Uniform Field
1 The equation for the torque
Refer to Figure 21.35(b). The magnitude of the torque τ acting on the 1-turn coil about the axis mentioned above and produced by the magnetic force F is given by
τ = F(a sin θ)
where a = AB. But the magnetic force F acting on BC (or AD) is given by
F = B I b
where b = BC (or AD). Hence
τ' = (BIb)(a sin θ)
; = BI(ab) sin θ
But, ab = area A of the plane of the coil
τ' = BIA sin θ
If the coil has N turns instead of one turn, the net torque τ would become
τ = Nτ'
τ = BINA sin θ
In general, the quantities B, I, N and A are kept constant. If this is the case, then we have
τ ∝ sin θ
In other words, as the torque forces the coil to rotate, hence causing angle θ to change, the magnitude of the torque would change accordingly. The torque does not remain constant as the coil rotates in the uniform magnetic field.
Note: (a) The angle θ is between the direction of the magnetic field and the normal to the plane of the coil. If we wish, we may use the angle between the direction of the field and the plane of the coil. Let this angle be α. Then we have the relationship
θ = 90° − α
sin θ = sin (90° − α) = cos α
The equation for torque would become
τ = BINA cos α
(b) In the discussion above, we consider a coil ABCD which has a rectangular shape. In general, the magnitude of the torque does not depend on the shape of the coil. The shape may be circular or oval. It is the area of the plane of the coil that counts.
2 Two particular cases
Consider the following two particular cases:
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| Figure 21.38 Illustration of a coil in a magnetic field at two positions: (a) θ = 0°, where the normal is parallel to the field and torque is zero, and (b) θ = 90°, where the normal is perpendicular to the field and torque is maximum (τ = BINA). |
(a) θ = 0°:
If θ = 0°, the normal is parallel to the direction of the field (Figure 21.38(a)). Then τ = 0.
(b) θ = 90°:
If θ = 90°, the normal is perpendicular to the field (Figure 21.38(b)). Then τ = BINA. This is the maximum torque obtainable.
21.7.3 Magnitude of the Torque in Radial Field
Figure 21.39 shows the plan view of the current-carrying rectangular coil ABCD mentioned above, which is now inside a radial magnetic field. The axis of rotation of the coil passes through the midpoint O, travels down the page and lies on the plane of the coil.
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Figure 21.39 Plan view of a current-carrying rectangular coil ABCD placed in a radial magnetic field, with the axis of rotation passing through the midpoint O along the plane of the coil. |
The field has the main feature that no matter what position the coil is in, we always find the plane of the coil parallel to the field. This feature is equivalent to the coil in a position which is parallel to the field, where θ = 90°, as shown in Figure 21.38(b).
When current I flows in the coil in the radial field, the torque produced by the current would be
τ = BINA sin 90°
or, τ = BINA
In other words, the magnitude of the torque remains the same even if the coil has rotated about the axis through any angle.
Note: If the coil is in the radial field as shown above, it would always be acted on by a torque of constant magnitude. Hence, it would rotate continuously as long as current flows in it. In other words, the whole setup now acts as a motor.
EXAMPLE 21.16
Refer to Figure 21.37. The rectangular coil has length 20 cm, breadth 15 cm and 20 turns. At places A and B, the magnetic field has strength 500 mT. If a current of 3.0 A flows in the coil, determine the magnitude of the torque that acts on the coil if the axis of rotation is at O.
Answer
The torque is given by
τ = BINA
= (0.50)(3.0)(20)(0.20 × 0.15)
= 0.90 N m
21.7.4 Moving-coil Galvanometer
1 Structure
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| Diagram showing the main components of a moving-coil galvanometer: a U-shaped permanent magnet producing a radial magnetic field, a soft iron core, a multi-turn rectangular coil mounted on bearings, a pair of control springs, and a pointer for indicating angular deflection. |
Figure 21.40 shows the components of a moving-coil galvanometer. The instrument has
(a) a U-shaped permanent magnet which provides a strong radial magnetic field.
(b) a small fixed soft iron cylinder placed in between the two magnetic poles.
(c) a rectangular coil of many turns which is supported on two bearings and with its plane parallel to the direction of the magnetic field.
(d) a pair of fine springs which are used to oppose the rotation of the coil
(e) a pointer which is attached to the coil and used for registering the angular deflection of the coil.
2 Principle
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| Figure 21.41 A moving-coil galvanometer showing the coil rotating due to magnetic torque and coming to rest when balanced by the opposing torque of the springs, producing an angular deflection α. |
(a) When current flows in the rectangular coil, a torque acts on the coil and tends to rotate the coil about the axis which passes through the two bearings.
(b) Since the plane of the coil is always parallel to the magnetic field, the magnitude of the torque τ is given by
τ = BINA
where B is the magnetic field strength, I the current flowing in the coil, N the number of turns of the coil, A the area of the plane of the coil.
(c) The pair of fine springs begins to unwind when the coil rotates. Eventually, the coil stops rotating when
torque τ acting on coil = opposing torque τ' produced by the springs
(d) But,; τ' ∝ α
where α is the angle turned through by the coil after the coil has come to rest, as shown in Figure 21.41.
τ' = kα
where k is the spring constant.
Hence, ; BINA = kα
Normally, the quantities B, N, A and k are kept constant. Then we have
I = (k / BNA) α
or, ; I ∝ α
This means that the angular displacement of the coil about its axis from its initial rest position is a measure of the current that flows through the coil.
EXAMPLE 21.17
The coil found inside a moving-coil galvanometer has dimensions of 2.0 cm × 2.0 cm and 50 turns. The diameter of the wire of the coil is 0.10 mm. The strength of the radial magnetic field at each vertical side of the coil is 500 mT. The spring that provides the counter torque to the coil has a spring constant of 2.0 × 10−5 N m per division of the scale of the galvanometer. A voltage of 52 mV is applied to the coil. Determine
(a) the resistance of the coil
(b) the torque that acts on the coil
(c) the number of scale divisions turned through by the pointer from its rest position.
(Resistivity of the material of the coil = 1.7 × 10−8 Ω m)
Answer
(a) Resistance R of the coil is given by
R = ρL / A1
Let a and b represent the length and breadth of the coil. Then we have
L = N(2a + 2b)
= (50)(2)(0.020 + 0.020) = 4.0 m
A1 = 1/4 πd2
= 1/4 π (0.10 × 10−3)2 = 7.85 × 10−9 m2
Hence,
R = (1.7 × 10−8)(4.0) / (7.85 × 10−9)
= 8.66 ≈ 8.7 Ω
(b) The current I which flows in the coil is given by
I = V / R
= (52 × 10−3) / 8.66
= 6.0 mA
The torque acting on the coil is given by
τ = BINA2
where A2 represents the area of the plane of the coil.
τ = (0.500)(6.0 × 10−3)(50)(0.020 × 0.020)
= 6.0 × 10−5 N m
(c) We have
I = (k / BNA2) α
or,
α = BINA2 / k
= (6.0 × 10−5) / (2.0 × 10−5)
= 3.0 divisions
EXERCISE 21.4
The coil of a moving-coil galvanometer has length 2.5 cm and breadth 1.5 cm, and has 50 turns. The strength of the radial magnetic field at the vertical side of the coil is 0.30 T. Determine the torque that acts on the coil when a current of 1.0 mA flows in the coil.
21.7.5 Simple D.C. Motor
1 Structure
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| Figure 21.42 A moving-coil galvanometer showing the coil rotating due to magnetic torque and coming to rest when balanced by the opposing torque of the springs, producing an angular deflection α. |
Figure 21.42 shows a diagram of the structure of a dc motor. The motor has the following components:
(a) There is a coil PQRS placed inside a magnetic field.
(b) The coil is wound round a cylindrical laminated iron core. The core together with the coil is referred to as the armature.
(c) The two free ends of the coil are connected to two parts, C1 and C2, of a split ring, known as a commutator.
(d) Each part of the ring makes contact with a conductor known as a brush, B1 or B2. The armature of the motor is connected to an external voltage supply via the brushes and the commutator.
2 Principle
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Figure 21.43 Diagram showing the interaction between the magnetic field produced by the current in coil PQRS and the external uniform magnetic field, resulting in a combined (resultant) magnetic field pattern. |
(a) Consider the coil PQRS placed in a horizontal position inside a uniform magnetic field, with PQ and RS perpendicular to the field. When current flows in the coil, the current will produce a magnetic field. The two magnetic fields interact to produce a resultant field, whose pattern is shown in Figure 21.43.
(b) Both PQ and RS are acted on by magnetic forces since current flows through them and they are inside a magnetic field.
(c) Suppose C1 and C2 are in contact with brushes B2 and B1 respectively, as shown in Figure 21.43(b). With this connection, current flows from P to Q and from R to S. We can use Fleming’s left hand rule to show that the direction of the magnetic force acting on PQ is upwards while the direction of the magnetic force acting on RS is downwards.
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| Figure 21.44 Diagram showing current flow in coil PQRS with commutator and brushes, where magnetic forces act upwards on PQ and downwards on RS according to Fleming’s left-hand rule. |
(d) The two forces are separated from and also parallel to each other. They have the same magnitude but are in opposite direction. Hence, they form a couple which tends to rotate the coil in the clockwise direction about an axis which passes through the coil and is perpendicular to the field.
(e) The angular momentum possessed by the rotating coil enables the coil to rotate through an angle which is greater than 90°. PQ, previously on the right hand side, has now moved to the left hand side, while RS has moved to the right hand side. This new position of the coil results in C1 (previously touching B2) making contact with B1 and C2 with B2.
(f) The current in the coil now flows in the opposite direction. Consequently, the magnetic force acting on PQ points upwards (previously pointing downwards) while that acting on RS points downwards. Hence, the torque acting on the coil still forces the coil to rotate in the clockwise direction.
(g) As PQ moves back to the right hand side again, the current changes direction. The magnetic force acting on PQ once again points downwards. Hence, the torque acting on the coil still forces the coil to rotate in the clockwise direction.
(h) With C1 and C2 making contact alternately with B1 and B2, the direction of the torque does not change at all. Consequently, the coil can be forced by the torque to rotate non-stop always in the same direction.
Applications of Torque on a Current-Carrying Coil
The principle of torque acting on a current-carrying coil has many important practical applications in electrical devices. Some common applications are described below:
1 Moving-coil Galvanometer
A moving-coil galvanometer uses the torque produced on a current-carrying coil placed in a magnetic field to measure small electric currents. The deflection of the coil is proportional to the current flowing through it, allowing accurate current measurement.
2 Electric Motor (D.C. Motor)
A D.C. motor operates based on the torque acting on a current-carrying coil in a magnetic field. The torque causes continuous rotation of the coil, converting electrical energy into mechanical energy. This principle is widely used in fans, mixers, and industrial machines.
3 Ammeters and Voltmeters
These instruments are developed from the moving-coil galvanometer. By modifying the galvanometer, it can measure larger currents (ammeter) or potential differences (voltmeter), using the same torque principle.
4 Loudspeakers
In loudspeakers, a coil carrying current is placed in a magnetic field. The torque and forces produced cause vibrations, which generate sound waves.
5 Measuring Instruments
Various analog measuring devices use the torque on a coil to produce pointer deflection, enabling the measurement of electrical quantities.
Conclusion
The torque on a current-carrying coil is produced due to the interaction between the magnetic field and the current flowing through the coil. This interaction generates a pair of forces that form a couple, causing the coil to rotate.
The magnitude of the torque depends on factors such as the magnetic field strength, current, number of turns, area of the coil, and the angle between the field and the normal to the coil. In a uniform magnetic field, the torque varies with angle, while in a radial magnetic field, the torque remains constant.
This principle is fundamental in many practical applications, including moving-coil galvanometers, ammeters, voltmeters, and electric motors, where electrical energy is converted into mechanical motion or used for precise measurement.
Overall, the concept of torque on a coil plays a crucial role in understanding electromagnetic devices and their real-world applications.
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