21.4 MAGNETIC FIELDS DUE TO CURRENTS
21.4.1 Magnetic Field Due to Current Flowing in Long Straight Conductor
1. Magnetic field pattern
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| Figure 21.15 The magnetic field produced by a steady current in a long, straight conductor forms concentric circular field lines around the wire. The direction of the magnetic field is given by the right-hand rule: the thumb points in the direction of the current, while the curled fingers indicate the direction of the magnetic field. |
Consider a very long, straight, thin conductor. A constant current flowing through it would produce a stable magnetic field in the free space surrounding the conductor. The pattern of the field consists of magnetic field lines which form concentric circles around the conductor, as shown in Figure 21.15.
2. Direction of magnetic field lines
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| Figure 21.16 The diagram shows how the right-hand rule is used to determine the direction of the magnetic field around a straight current-carrying conductor. The thumb points in the direction of the current, while the curled fingers indicate the direction of the circular magnetic field lines surrounding the conductor. |
We can use the right-hand rule to assist us in determining the field direction. We use the rule in the following manner:
(a) Point the thumb in the direction of the current.
(b) Use all the fingers of the right hand to wrap around a portion of the conductor. Then each finger would be pointing in the direction of the field, as shown in Figure 21.16.
3. Magnitude of magnetic field strength
Suppose that a current I flows in the long, straight conductor placed in free space. Then the magnitude of the magnetic field strength B at a point at perpendicular distance r from the conductor is given by
$\boxed{B = \frac{\mu_0I}{2\pi r}}$
where μ0 is a constant known as the permeability of free space. The constant has a value of 4π × 10−7 H m−1. Notice that we have the relationships
$B ∝ I$
$B ∝ \frac{1}{r}$
Note (a) This formula is applicable only if the conductor is
(i) very long (theoretically it should be infinitely long)
(ii) straight
(iii) thin so that its diameter is negligible and
(iv) placed in free space, i.e., without any materials around the conductor to influence the magnetic field produced by the current.
(b) H in the unit H m−1 represents the unit henry.
EXAMPLE 21.4
A current of 5.0 A flows in a long straight conductor in free space. The Earth’s magnetic field may be neglected. Determine the magnetic field strength at a point which is at a perpendicular distance of 2.0 cm from the conductor.
Answer
At the point stated,
$B = \frac{\mu_0I}{2\pi r}$
$= \frac{(4\pi \times 10^{-7})(5.0)}{2\pi(2.0 \times 10^{-2})}$
$= 5.0 \times 10^{-5} \ T$
EXAMPLE 21.5
The magnetic field strength at a point 3.0 cm from a long, thin, straight conductor which carries a current of 6.0 A is x T. Determine the magnetic field strength, in terms of x, at another point 4.5 cm away from the conductor which now carries a current of 12 A.
ANSWER
For the first case,
$B_1 =k\frac{I_1}{r_1}=x$
For the second case,
$B_2 =k\frac{I_2}{r_2}$
$\frac{B_1}{B_2}=\left(\frac{I_1}{I_2}\right)\left(\frac{r_2}{r_1}\right)$
$\frac{x}{B_2}=\left(\frac{6.0}{12}\right)\left(\frac{4.5}{3.0}\right)$
$B_2 = 1.3x \ T$
EXAMPLE 21.6
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| Figure 21.17 The neutral point is the position where the magnetic field of the wire cancels Earth’s magnetic field, resulting in zero net field. |
Answer
(a) The magnetic field produced by the current is superposed on the Earth’s magnetic field, as shown in Figure 21.18.
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Figure 21.18 The magnetic field produced by the current combines with Earth’s magnetic field, resulting in a modified overall field pattern. |
(b) At point X, the two fields have the same strength but oppose each other. In other words, the distance of X from the wire is such that the strength BI of the field produced by the current is equal in magnitude to the strength BE of the Earth’s field.
$B_I=B_E$
$\frac{\mu_0I}{2\pi r} = 20 \ \mu T$
$r = \frac{(4 \pi \times 10^{-7})(10.0)}{2 \pi (20 \times 10^{-6})}$
$r= 0.10 \ m$
X is on the west side of the wire.
EXAMPLE 21.7
A long, straight current-carrying wire is placed inside a uniform magnetic field. The wire is perpendicular to the direction of the field. Draw a pattern of the magnetic field which results from the interaction between the magnetic field produced by the current and the given field. Show the direction of the magnetic force which acts on the wire.
Answer
Figure 21.19 shows how the resultant magnetic field looks like.
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| Figure 21.19 The diagram shows the combined magnetic field formed by the interaction of the wire’s magnetic field and the uniform external field. The distortion of field lines indicates the direction of the magnetic force acting on the wire. |
21.4.3 Magnetic Field Due to Current Flowing in Circular Coil
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Figure 21.20 The diagram shows the magnetic field produced by a circular coil, with field lines indicating its direction. The direction of the field reverses when the direction of the current is reversed. |
1. Magnetic field pattern
Figure 21.20 shows how the pattern of the magnetic field produced by the current flowing in the circular coil looks like. It also shows the direction of the field. The direction would be reversed if the current is reversed.
2. Magnitude of magnetic field strength at the centre of the coil
Consider a circular coil in free space which has N turns, radius R and carries a current I. Then the magnitude B of the magnetic field strength at the centre of the coil is given by
$\boxed{B=\frac{\mu_0NI}{2R}}$
Notice that we have the relationships
EXAMPLE 21.8
A circular coil P carrying a current of 5.0 A has a radius of 3.0 cm and 50 turns. Another circular coil Q carrying a current of 8.0 A has a radius of 4.0 cm and 40 turns. Determine the ratio of the magnetic field strength at the centre of P to that at the centre of Q.
Answer
At the centre of coil P,
$B_P=\frac{\mu_0N_PI_P}{2R_P}$
At the centre of Q,
$B_Q=\frac{\mu_0N_QI_Q}{2R_Q}$
$\frac{B_P}{B_Q} =\left(\frac{N_P}{N_Q}\right)\left(\frac{I_P}{I_Q}\right)\left(\frac{R_Q}{R_P}\right)$
$\frac{B_P}{B_Q} = (50 / 40) (5.0 / 8.0) (4.0 / 3.0)$
$\frac{B_P}{B_Q} = 1.04$
EXAMPLE 21.9
A circular coil of radius 2.0 cm has 100 turns. Its plane is kept in a vertical position and a small magnetic compass is placed at its centre. The plane of the coil is adjusted until it is parallel to the compass needle. Figure 21.21 shows a plan view of the cross section of the coil and the direction of current flow. The strength of the horizontal component of the Earth’s magnetic field is 20 μT. A current of 0.50 A is allowed to flow in the coil. Determine the deflection of the compass needle.
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Figure 21.21 The diagram shows a circular coil placed parallel to a compass needle. When current flows through the coil, its magnetic field interacts with Earth’s magnetic field, causing the compass needle to deflect. |
Answer
A resultant magnetic field of strength B exists at the centre of the coil. It is produced by the interaction of the following two magnetic fields:
(a) the field of strength BC produced by the current flowing in the coil
(b) the horizontal component of strength BE of the Earth’s magnetic field
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| Figure 21.22 The diagram shows the vector addition of the magnetic field produced by the coil (BC) and the Earth’s horizontal magnetic field (BE), resulting in a resultant magnetic field that determines the direction of the compass needle. |
Figure 21.22 shows how the two field vectors are added to produce the resultant field vector. Referring to the vector diagram, we get
$tan \ \theta = \frac{B_C}{B_E}$
$= \left(\frac{1}{B_E}\right) \left(\frac{\mu_0NI}{2R}\right)$
$=\frac{(4\pi \times 10^{-7})(100)(5.00 \times 10^{-2})}{(2)(20 \times 10^{-6})(2.0 \times 10^{-2})} $
$tan \ \theta = 7.854$
$\theta = tan^{-1}(7.854)=82.7^0$ to the Earth’s magnetic field
Applications of Magnetic Fields Due to Electric Currents
Magnetic fields produced by electric currents have many important applications in daily life, industry, and technology. These applications are based on the principles of magnetic fields around straight conductors and circular coils.
1. Electric Motors
Electric motors use the interaction between a magnetic field and a current-carrying conductor to produce motion. When current flows through a coil placed in a magnetic field, a force acts on the coil, causing it to rotate. This principle is widely used in fans, washing machines, and electric vehicles.
2. Electromagnets
A current flowing through a coil produces a strong magnetic field, turning the coil into an electromagnet. The strength of the magnetic field can be increased by increasing the current or the number of turns in the coil. Electromagnets are used in cranes to lift heavy metal objects and in electric bells.
3. Magnetic Compass Deflection
The magnetic field produced by a current can affect a compass needle. This principle is used in devices such as galvanometers to detect and measure small electric currents based on the deflection of a needle.
4. Magnetic Field Measurement
The relationship between current and magnetic field strength allows us to calculate unknown currents or distances. This principle is applied in sensors and measuring instruments used in laboratories and engineering.
5. Transmission of Electric Power
Current flowing in power lines produces magnetic fields around the conductors. Understanding these fields helps in designing safe and efficient power transmission systems.
6. Medical Applications
Magnetic fields generated by coils are used in medical technologies such as MRI (Magnetic Resonance Imaging), where strong and controlled magnetic fields are required to produce detailed images of the human body.
7. Induction Devices
Changing magnetic fields produced by currents in coils can induce currents in nearby conductors. This principle is used in transformers, wireless charging systems, and induction cookers.
Conclusion
Magnetic fields due to electric currents are fundamental in understanding how electricity and magnetism interact. A current flowing through a long straight conductor produces circular magnetic field lines, while a circular coil generates a stronger and more concentrated magnetic field at its centre. The strength of these fields depends on factors such as current, distance, number of turns, and coil radius.
The concept of superposition explains how magnetic fields combine, leading to phenomena such as neutral points and compass deflection. These principles are essential in explaining real-world applications and form the basis for many electrical and electromagnetic devices.
Frequently Asked Questions (FAQ)
1. What is a magnetic field?
A magnetic field is a region where a magnetic force can be experienced by a moving charge or magnetic material.
2. What produces a magnetic field?
Magnetic fields are produced by moving electric charges or electric currents.
3. What is the pattern of a magnetic field around a straight wire?
It forms concentric circular field lines around the wire.
4. How do you determine the direction of the magnetic field?
By using the right-hand rule.
5. What is the right-hand rule?
A rule where the thumb shows current direction and curled fingers show magnetic field direction.
6. What factors affect magnetic field strength in a straight conductor?
Current and distance from the conductor.
7. What is the formula for magnetic field around a straight wire?
B = μ₀I / 2πr.
8. What is a neutral point?
A point where the net magnetic field is zero due to equal and opposite fields.
9. What happens when magnetic fields overlap?
They combine according to the principle of superposition.
10. How does a circular coil produce a magnetic field?
It produces a magnetic field similar to a bar magnet, concentrated at the center.
11. What affects the magnetic field of a circular coil?
Number of turns, current, and radius of the coil.
12. What is the formula for the magnetic field at the center of a coil?
B = μ₀NI / 2R.
13. Why does a compass deflect near a current-carrying wire?
Because the magnetic field from the current interacts with Earth’s magnetic field.
14. What happens if the current direction is reversed?
The direction of the magnetic field also reverses.
15. Why is this topic important?
It is essential for understanding electrical devices like motors, generators, and electromagnets.
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