21.5 FORCE BETWEEN CURRENT-CARRYING CONDUCTORS
21.5.1 Magnetic Force between Two Long, Straight, Thin, Parallel Current-carrying Conductors
1. Currents in same direction
(a) Consider two infinitely long, straight, thin conductors in free space which are parallel to each other. Current flows in each conductor and the direction of current flow in one conductor is the same as that in the other conductor, as shown in Figure 21.27(a). Then the conductors in this arrangement will attract each other by magnetic force.
(b) Each current produces a magnetic field. The two fields interact with each other and produce a resultant field. The pattern of this resultant field, without taking into account the Earth’s magnetic field, is shown in Figure 21.27(b).
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| Figure 21.27 (a) Two long, straight, parallel conductors carry currents in the same direction, causing an attractive magnetic force between them. (b) The magnetic fields produced by each conductor interact to form a resultant magnetic field pattern around the conductors, as shown, without considering the Earth’s magnetic field. |
2. Currents in opposite direction
(a)
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| Figure 21.28 (a) Two long, straight, parallel conductors carry currents in opposite directions, causing a repulsive magnetic force between them. (b) The magnetic fields produced by each conductor interact to form a resultant magnetic field pattern around the conductors, as shown, without considering the Earth’s magnetic field. |
Suppose the direction of current in one conductor is opposite to the direction of current in the other conductor, as shown in Figure 21.28(a). Then the conductors will repel each other by magnetic force.
(b) The pattern of the resultant field, without taking into account the Earth’s magnetic field, is shown in Figure 21.28(b).
3. Magnitude of attractive or repulsive magnetic force
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| Figure 21.29 Two long, straight, parallel conductors AB and CD are separated by a distance d. |
Consider two long thin straight parallel conductors, AB and CD, which are separated by distance d, as shown in Figure 21.29. AB carries current I1 which flows upwards, and CD carries current I2 which also flows upwards. The magnetic field strength B produced by current I1 at the position of the conductor CD is given by
B = μ0I1 / (2πd)
The current I2 in CD is found inside this magnetic field. Hence, a magnetic force F acts on any length L of the conductor CD. The magnitude of this force is given by
F = BI2L
= ( μ0I1 / 2πd ) I2L
The force acting on unit length of the conductor CD is given by
F / L = μ0I1I2 / 2πd
Using the same method, we can show that the magnitude of the magnetic force acting on unit length of the conductor AB is also given by the above equation. But the direction of the force is opposite to that of the force acting on CD. This means that the magnetic force acting on CD (or AB) is an action and the force acting on AB (or CD) is a reaction. These two forces oppose each other but are equal in magnitude, in accordance with Newton’s third law of motion.
EXAMPLE 21.13
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| Figure 21.30 Three long, straight, parallel conductors P, Q, and R lie on the same plane. |
Three long, straight, thin conductors, P, Q, R, lie on the same plane and are parallel to each other, as shown in Figure 21.30. Current flows in each conductor in the direction as shown. The Earth’s magnetic field may be neglected. The resultant magnetic force acting on Q is 0.50 mN m-1 pointing towards P.
(a) Is the direction of the current in R shown in the diagram correct? Explain.
(b) Determine the current in R.
Answer
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| Figure 21.31 A plan view showing the cross-section of three parallel conductors P, Q, and R. |
(a) (i) Figure 21.31 shows a plan view of the cross-section of the three conductors.
(ii) Current IP in P produces magnetic field of strength BP. Since conductor Q carries a current and lies inside this field, it will be acted on by a magnetic force FP. We can use Fleming’s left hand rule to confirm that FP points towards R, as shown.
(iii) Current IR in conductor R produces magnetic field of strength BR. Since conductor Q also lies inside this field, it will be acted on by a second magnetic force, FR.
(iv) It is given that the resultant force acting on Q points towards P. Hence, we must have FR pointing towards P and also FR > FP in order to have the resultant force pointing towards P.
(v) In order to have force FR pointing towards P, the current in R must flow upwards. Hence, the direction of current flow shown in Figure 21.30 is correct.
(b)
FR / L − FP / L = 0.50 mN m-1
But,
FR / L = μ0IQIR / 2πdQR
= (4π × 10-7)(5)IR / 2π(0.02)
= 5.0 × 10-5IR N m-1
FP / L = μ0IPIQ / 2πdPQ
= (4π × 10-7)(10)(5) / 2π(0.04)
= 25.0 × 10-5 N m-1
Hence 5.0 × 10-5IR − 25.0 × 10-5 = 0.50 × 10-3
÷ 5.0 × 10-5, IR − 5 = 10
IR = 15 A
EXAMPLE 21.14
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| Figure 21.32 A long straight wire XY carries a current of 5.0 A, while a rectangular coil PQRS with 10 turns carries a current of 3.0 A. The side PS is placed parallel to the wire at a distance of 2.0 cm. |
Current of 5.0 A flows in a long straight vertical wire, XY, and current of 3.0 A flows in a rectangular coil, PQRS. The coil has 10 turns and dimensions PQ = 3.0 cm and PS = 10 cm. The side PS is placed parallel and 2.0 cm away from XY, as shown in Figure 21.32.
(a) Determine the magnitude and direction of the magnetic force that acts on
(i) PS (ii) QR.
(b) Will the coil rotate about an axis? Explain.
Answer
(a)
(i) The strength BPS at a point on PS of the magnetic field produced by the current IXY in XY is given by
BPS = μ0IXY / 2πr1
= (4π × 10-7)(5.0) / 2π(2.0 × 10-2) = 5.0 × 10-5 T
The magnetic force FPS acting on PS is given by
FPS = NBPSIPSℓPS(PS) sin 90°
= (10)(5.0 × 10-5)(3.0)(10 × 10-2) = 1.5 × 10-4 N
The force (i) lies on the plane of PQRS (ii) is perpendicular to PS and (iii) points towards XY.
(ii) The strength BQR at a point on QR of the magnetic field produced by the current IXY in XY is given by
BQR = μ0IXY / 2πr2
= (4π × 10-7)(5.0) / 2π(5.0 × 10-2) = 2.0 × 10-5 T
The magnetic force FQR acting on QR is given by
FQR = NBQRIQRℓQR(QR) sin 90°
= (10)(2.0 × 10-5)(3.0)(10 × 10-2) = 0.6 × 10-4 N
The force (i) lies on the plane of PQRS (ii) is perpendicular to QR and (iii) points away from XY.
(b) The magnetic forces FPS and FQR lie on the same plane. Hence, they do not produce a torque on the coil.
The magnetic forces acting on PQ and RS both lie on the plane of PQRS. Hence, they too do not produce a torque on the coil. The coil will not rotate. Instead, it will move towards XY, if it is allowed to do so.
Applications of Magnetic Force Between Current-Carrying Conductors
The magnetic force between current-carrying conductors has many important applications in electrical and electronic systems. This principle is widely used in modern technology to control motion, generate force, and transfer energy efficiently.
1. Electric Motors
The interaction between magnetic fields and current-carrying conductors is the basic working principle of electric motors. Forces produced cause rotation, converting electrical energy into mechanical energy.
2. Electromagnets
Parallel conductors carrying current can enhance magnetic effects, which are used in electromagnets for lifting heavy objects, magnetic separation, and industrial machinery.
3. Power Transmission Lines
In high-voltage transmission lines, currents flowing in parallel wires produce magnetic forces. These forces must be considered in the design to prevent wires from moving or colliding.
4. Railguns and Magnetic Launch Systems
Strong magnetic forces between conductors are used to accelerate objects at high speeds in railgun technology and electromagnetic launch systems.
5. Measuring Instruments
Devices such as galvanometers and ammeters use magnetic forces on current-carrying conductors to measure electric current accurately.
6. Magnetic Levitation (Maglev)
Magnetic forces between currents can be used to produce lift, as seen in maglev trains, reducing friction and allowing high-speed transportation.
These applications show how the interaction between electric current and magnetic fields plays a crucial role in modern science and engineering.
Conclusion
The magnetic force between current-carrying conductors is a fundamental concept in electromagnetism. When currents flow in the same direction, the conductors attract each other, while currents in opposite directions cause repulsion. This interaction is explained by the magnetic fields produced around each conductor.
The magnitude of the force depends on factors such as the currents, the distance between the conductors, and the length of the wires. Understanding these relationships allows accurate prediction and control of magnetic effects in practical systems.
This principle is widely applied in technologies such as electric motors, power transmission systems, and electromagnetic devices. Therefore, studying magnetic forces between conductors is essential for understanding and developing modern electrical and engineering applications.
Frequently Asked Questions (FAQ)
1. What is magnetic force between current-carrying conductors?
It is the force that arises due to the interaction of magnetic fields produced by currents flowing in nearby conductors.
2. When do conductors attract each other?
Two conductors attract each other when the currents flowing through them are in the same direction.
3. When do conductors repel each other?
Two conductors repel each other when the currents flowing through them are in opposite directions.
4. What factors affect the magnitude of the magnetic force?
The magnitude of the force depends on the currents in the conductors, the distance between them, and the length of the conductors.
5. What is the formula for magnetic force per unit length?
The magnetic force per unit length between two parallel conductors is given by F/L = μ₀I₁I₂ / (2πd).
6. Why is the force between conductors important?
This force is important because it forms the basis of many applications such as electric motors, electromagnets, and power transmission systems.
7. How is the direction of the force determined?
The direction of the force can be determined using Fleming’s left-hand rule or by analyzing the direction of the magnetic fields around the conductors.
8. Is this principle used in real-life applications?
Yes, it is widely used in electrical devices, industrial systems, and modern technologies such as magnetic levitation and electromagnetic machinery.
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