21.5 SOLENOIDS AND TOROIDS
21.5.1 Solenoids
1. What is a Solenoid?
A solenoid is a long coil of wire that is tightly wound in the shape of a cylinder. When an electric current flows through the wire, it produces a magnetic field around the coil.
A long solenoid behaves like a bar magnet. One end acts as a north pole, while the other end acts as a south pole. The magnetic field inside the solenoid is strong and uniform, whereas the field outside is weak.
The strength of the magnetic field inside a solenoid depends on:
- The current flowing through the wire
- The number of turns per unit length
- The type of core material inside the solenoid (e.g., air or iron)
Solenoids are widely used in devices such as electric bells, relays, and electromagnetic locks because they can convert electrical energy into magnetic energy.
2. Is There Metal in a Solenoid?
Yes, a solenoid can have metal—but it is not always required, as show in Figure 21.23.
Here’s the explanation:
Air-core solenoid:
- No metal inside the coil.
- It consists only of a wound wire (usually copper).
- A magnetic field is produced, but it is relatively weak.
Iron-core solenoid:
- Contains a metal core, usually soft iron, inside the coil.
- The metal core greatly strengthens the magnetic field.
- Commonly used in devices like electromagnets, relays, and electric bells.
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| Figure 21.23 (a) Air-core solenoid: A coil of wire without any metal inside. The magnetic field is relatively weak, depending only on the current and number of turns. (b) Iron-core solenoid: A coil with a soft iron core inside. The metal core strengthens the magnetic field, making the solenoid more effective as an electromagnet. |
3. How Does a Solenoid Work?
A solenoid works based on the principle that an electric current flowing through a conductor produces a magnetic field. When the wire is wound into a coil, the magnetic fields from each loop combine to form a stronger, uniform magnetic field inside the solenoid.
When current flows through the solenoid:
- Each turn of the coil produces a circular magnetic field.
- These magnetic fields add together inside the coil, creating a strong and nearly uniform magnetic field.
- The solenoid behaves like a bar magnet, with one end acting as a north pole and the other as a south pole.
The direction of the magnetic field depends on the direction of the current. This can be determined using the right-hand grip rule: if you curl your fingers in the direction of current flow, your thumb points in the direction of the magnetic field (north pole).
When the current is switched off, the magnetic field disappears, making the solenoid useful for applications that require controlled magnetism.
21.5.2 Magnetic Field Due to current Flowing in Long Solenoid
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| Figure 21.24 The figure shows the magnetic field pattern of a long solenoid, where the field inside is strong and uniform, while the field outside resembles that of a bar magnet. |
1. Magnetic field pattern
Consider a solenoid which is very long and having a diameter which is relatively small compared with the length. When a current flows in it a magnetic field would be produced. The pattern of the field is shown in Figure 21.24. The field pattern outside the solenoid is very similar to that produced by a bar magnet.
2 Direction of magnetic force lines
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| Figure 21.25 Clockwise current indicates a south pole, while anticlockwise current indicates a north pole at the end of the solenoid |
If we view the solenoid through one end of the solenoid, we would ‘see’ the current flowing either in the clockwise or anti-clockwise direction, as shown in Figure 21.25. The direction of current flow will determine whether the viewed end of the solenoid acts as a magnetic north pole or a magnetic south pole. If the end acts as a north pole, the magnetic force lines would point outwards, away from the solenoid.
3. Right-Hand Thumb Rule for Solenoids
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| Figure 21.26 The image illustrates the right-hand rule for a solenoid. When you curl the fingers of your right hand in the direction of the current flowing through the solenoid coils, your thumb points in the direction of the magnetic field inside the solenoid. |
The right-hand thumb rule is used to determine the direction of the magnetic field inside a solenoid, as shown in Figure 21.26.
To apply this rule:
- Hold the solenoid in your right hand.
- Wrap your fingers around the coil in the direction of the current flow.
- Your thumb will point in the direction of the magnetic field inside the solenoid.
The direction in which your thumb points indicates the north pole of the solenoid.
If the current direction is reversed, the direction of the magnetic field also reverses, and the north and south poles swap positions.
This rule helps in understanding how solenoids behave like magnets and is important in applications such as electromagnets and electric devices.
4. Magnitude of magnetic field strength
Consider a long solenoid which has n turns per unit length, $N/L$. A current I flows through the solenoid. Then the magnetic field strength B at any point inside the solenoid and far from each end
(a) is constant in magnitude
(b) has a magnitude of
$\boxed{B=\mu_0nI=\frac{\mu_0NI}{L}}$
where:
B = magnetic field strength
μ₀ = permeability of free space
n = number of turns per unit length
L = circumference of the circular path in the toroid
I = current flowing through the coil
Figure 21.27 shows how the magnetic field strength B varies with the position of a point that lies on the axis of the solenoid.
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| Figure 21.27 The graph shows how the magnetic field strength (B) varies along the axis of a solenoid, remaining nearly constant at the center and decreasing toward the ends. |
21.5.2 Toroids
1. What is a Toroid?
A toroid is a coil of wire wound into a circular ring (doughnut-shaped) form.
It can be considered as a solenoid that is bent into a closed loop, where the magnetic field is confined within the core, resulting in a stronger magnetic field and minimal leakage outside the coil.
2. Is There Metal in a Toroid?
A toroid may or may not contain metal, depending on its design and application.
Air-core toroid: No metal core is used. The magnetic field is produced only by the current in the coil, but it is relatively weaker.
Iron-core toroid: A metal core, usually soft iron or ferrite, is placed inside the coil. This greatly strengthens the magnetic field and improves efficiency.
Therefore, metal is not always required, but it is commonly used to enhance the magnetic field in practical applications.
3. Right-Hand Thumb Rule for Solenoids
The right-hand thumb rule can also be applied to a toroid to determine the direction of the magnetic field.
When the fingers of the right hand are curled in the direction of the current flowing through the toroid, the thumb indicates the direction of the magnetic field inside the core.
The magnetic field forms closed circular paths within the toroid.
4. Magnitude of Magnetic Field Strength
The magnitude of the magnetic field strength inside a toroid depends on the current flowing through the coil and the number of turns.
The magnetic field inside a toroid is given by:
$\boxed{B=\mu_0nI=\frac{\mu_0NI}{L}=\frac{\mu_0NI}{2\pi r}}$
where:
B = magnetic field strength
μ₀ = permeability of free space
n = number of turns per unit length
L = circumference of the circular path in the toroid = $2\pi r$
I = current flowing through the coil
r = is the radius of the circular path inside the toroid
The magnetic field is strongest inside the core and decreases as the distance from the center increases.
EXAMPLE 21.10
A current of 5.0 A flows in a solenoid of length 2.0 m. The diameter of the solenoid is small compared with the length of the solenoid. Determine the number of turns which the solenoid must have in order to obtain a magnetic field strength of 2.0 mT at a point inside the solenoid and far from each end of the solenoid.
Answer
Given:
Current, $I = 5.0 \ A$
Length of solenoid, $L = 2.0 \ m$
Magnetic field strength, $B = 2.0 \times 10^{-3} \ T$
Permeability of free space, $\mu_0= 4\pi \times 10^{-7} \ T \ m \ A^{-1}$
Formula:
$B = \mu_0nI$
$n = \frac{B}{\mu_0I}$
$n = \frac{2.0 \times 10^{-3}}{(4\pi \times 10^{-7})(5.0)}$
$n = 3.18 \times 10^2 \ m^{-1}$
Number of turns:
$N = n \times L$
$N = (3.18 × 10^2)(2.0)$
$N = 636$
EXAMPLE 21.11
A copper wire of uniform diameter 1.0 mm is wound round a cardboard tube, which has a small diameter, so that a long solenoid is formed. The wire is wound in such a way that the turns touch each other. If a current of 2.0 A flows through the solenoid, determine the magnetic field strength at a point inside and near the middle section of the solenoid.
Answer
Let t = thickness of wire = 1.0 mm
N = number of turns found in length L of solenoid
The magnetic field strength B is given by
B = μ₀ n I
But, $n = N/L$
and $L = Nt$
Hence, $B = \mu_0 \left(\frac{1}{t}\right) I$
$= (4\pi \times 10^{-7}) \left(\frac{1}{1.0 \times 10^{-3}}\right) (2.0)$
$= 2.5 \times 10^{-3}\ T$
EXAMPLE 21.12
A circular coil and a long solenoid are arranged in the way as shown in Figure 21.28. The axis of the solenoid passes through the centre of the coil and is perpendicular to the plane of the coil. The coil has 10 turns, a diameter of 5.0 cm and carries a current of 5.0 A. The solenoid has 1 000 turns per metre and carries a current of 0.10 A. Each arrow in the diagram indicates the direction of current flow. Determine the magnitude of the magnetic field strength at the centre of the coil.
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| Figure 21.28 A circular coil is placed around a long solenoid such that the axis of the solenoid passes through the center of the coil and is perpendicular to the plane of the coil. This configuration is commonly used to study electromagnetic induction, where a changing magnetic field inside the solenoid induces an electric current in the circular coil. |
Answer
At the centre of the coil, the current Ic in the coil produces a magnetic field of strength Bc. The field points to the right.
$B_c=\frac{\mu_0I_c}{2R}$
$= \frac{(4\pi \times 10^{-7})(5.0)}{2(2.5 \times 10^{-2})}$
$= 1.26 \times 10^{-4} \ T$ to the right
At a point on the axis of the solenoid, the current Is in the solenoid produces a magnetic field of strength Bs. The field points to the right.
$B_s=\mu_0 n I$
$= (4\pi \times 10^{-7})(1000)(0.10)$
$= 1.26 \times 10^{-4} \ T$ to the right
The resultant magnetic field strength B at the centre of the coil is given by
$B = B_c+B_s$
$= (1.26 + 1.26) \times 10^{-4}$
$= 2.5 \times 10^{-4}\ T$ to the right
Figure 21.29 shows that the direction of the total magnetic field is to the right, indicating that the combination of the solenoid and the circular coil produces a stronger magnetic field.
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| Figure 21.29 shows that the total magnetic field is directed to the right. This indicates that the magnetic fields produced by the solenoid and the circular coil reinforce each other, resulting in a stronger combined magnetic field. |
EXAMPLE 21.13
A toroid having a square cross section, 5.41 cm on a side, and an inner radius of 12.3 cm has 670 turns and carries a current of 1.02 A. (It is made up of a square solenoid instead of a round one bent into a doughnut shape.) What is the magnitude of the magnetic field inside the toroid at:
(a) the inner radius
(b) the outer radius?
Answer
The magnetic field inside a toroid is given by:
$B=\frac{\mu_0NI}{2\pi r}$
Where:
$\mu_0 = 4\pi \times 10^{-7} \ T·m/A$
$N = 670 \ turns$
$I = 1.02 \ A$
r = radius (in meters)
(a) Magnetic Field at Inner Radius
Inner radius: $r_{in} = 12.3 \ cm = 0.123 \ m$
$B_{in}= \frac{4\pi \times 10^{-7} \times 670 \times 1.02}{2\pi \times 0.123}$
$B_{in}= \frac{4 \times 10^{-7} \times 670 \times 1.02}{2 \times 0.123}$
$B_{in} ≈ \frac{2.7336 \times 10^{-4}}{0.246}$
$B_{in} ≈ 1.11 \times 10^{-3} \ T$
(b) Magnetic Field at Outer Radius
Outer radius: $r_{out}= 12.3 \ cm + 5.41 \ cm = 17.71 \ cm = 0.1771 \ m$
$B_{out} = \frac{4\pi \times 10^{-7} \times 670 \times 1.02}{2\pi \times 0.1771}$
$B_{out}= \frac{2.7336 \times 10^{-4}}{0.3542}$
$B_{out} ≈ 7.72 \times 10^{-4} \ T$
Final Answers
(a) Magnetic field at inner radius = 1.11 × 10-3 T
(b) Magnetic field at outer radius = 7.72 × 10-4 T
EXAMPLE 21.14
A long solenoid has 92 turns/cm and carries a current i. An electron moves within the solenoid in a circle of radius 2.97 cm perpendicular to the solenoid axis. The speed of the electron is 0.0668c (where c = speed of light = 2.998 × 108 m/s). Find the current i in the solenoid.
Answer
Step 1: Magnetic field provides centripetal force
Magnetic force = centripetal force:
$F_L=F_c$
Simplify:
$B = \frac{mv}{qr}$
Step 2: Magnetic field inside a solenoid
$B=\mu_0 n i$
Where:
$\mu_0 = 4\pi \times 10^{-7} \ T·m/A$
n = number of turns per meter
Convert turns/cm to turns/m:
n = 92 turns/cm = 9200 turns/m
Step 3: Equate both expressions for B
$\mu_0 n i = \frac{mv}{qr}$
Solve for current:
$i = \frac{mv}{\mu_0 n q r}$
Step 4: Substitute values
$m = 9.11 \times 10^{-31} \ kg$
$q = 1.60 \times 10^{-19} \ C$
$r = 2.97 \ cm = 0.0297 \ m$
$v = 0.0668 \times 2.998 \times 10^8 = 2.00 \times 10^7 \ m/s$
$\mu_0 = 4\pi \times 10^{-7}$
$n = 9200$
$i = \frac{(9.11 \times 10^{-31})(2.00\times 10^7)}{(4\pi \times 10^{-7})(9200)(1.60 \times 10^{-19})(0.0297)}$
$i ≈ \frac{1.822 \times 10^{-23}}{5.49 \times 10^{-23}}$
$i ≈ 0.332 \ A$
The current in the solenoid is:
i ≈ 0.33 A
Applications of Solenoids and Toroids
Solenoids and toroids are important components widely used in electrical and electronic systems due to their ability to produce and control magnetic fields efficiently.
Solenoids
A solenoid is a long coil of wire that generates a nearly uniform magnetic field when an electric current flows through it. Solenoids are commonly used in:
- Electromagnets
- Electric relays
- Solenoid valves (used to control fluid flow)
- Inductors in electronic circuits
- Magnetic switches and actuators
Toroids
A toroid is a coil wound into a circular ring shape. It confines the magnetic field within its core, reducing energy loss and electromagnetic interference. Toroids are commonly used in:
- Transformers
- Inductors
- Power supplies
- Magnetic energy storage devices
- Noise filters in electronic circuits
Both solenoids and toroids play a crucial role in modern technology by improving efficiency, minimizing energy loss, and enabling precise control of magnetic fields.
Conclusion
Solenoids and toroids are essential components in the study and application of magnetic fields produced by electric currents. A solenoid generates a strong and nearly uniform magnetic field along its axis, making it highly useful in devices that require controlled magnetic effects. On the other hand, a toroid confines the magnetic field within its core, resulting in greater efficiency and minimal energy loss.
Their practical applications in devices such as electromagnets, transformers, inductors, and electronic circuits demonstrate their importance in modern technology. Understanding how solenoids and toroids work helps in designing efficient electrical systems and improving the performance of various electromagnetic devices.
Frequently Asked Questions (FAQ)
1. What is a solenoid?
A solenoid is a long coil of wire that produces a magnetic field when an electric current flows through it. The magnetic field inside a solenoid is strong and nearly uniform.
2. What is a toroid?
A toroid is a coil of wire wound into a circular (ring) shape. It confines the magnetic field within its core, making it more efficient and reducing energy loss.
3. What is the main difference between a solenoid and a toroid?
The main difference is that a solenoid produces a magnetic field that extends outside the coil, while a toroid confines the magnetic field within its core.
4. Why are toroids more efficient than solenoids?
Toroids are more efficient because their closed-loop shape prevents magnetic field leakage, reducing energy loss and electromagnetic interference.
5. What are the common applications of solenoids?
Solenoids are used in electromagnets, relays, solenoid valves, inductors, and various switching devices.
6. What are the common applications of toroids?
Toroids are commonly used in transformers, inductors, power supplies, and noise filters in electronic circuits.
7. How does current affect the magnetic field in a solenoid?
The strength of the magnetic field in a solenoid increases as the electric current increases.
8. Why is the magnetic field inside a solenoid uniform?
The magnetic field inside a solenoid is uniform because the field lines are parallel and evenly spaced along its length, especially in the central region.
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