Faraday’S Law Of Induction- Mutual And Self-Inductance - Example 3(Long And Short Coaxial Solenoid)

Faraday’s Law of Induction- Mutual and Self-Inductance - Example 3 (Long and Short Coaxial Solenoid)

In this example, we will analyze the mutual and self-inductance of a long and a short coaxial solenoid.

Slide 2:

A long solenoid has length L, whereas a short solenoid has length much smaller than the radius of the long solenoid.
The long solenoid has N1 turns per unit length, while the short solenoid has N2 turns per unit length.

Slide 3:

Let’s start with the long solenoid. The self-inductance of the long solenoid is given by the formula: L1 = μ₀N₁²A/L
Where μ₀ is the permeability of free space, N₁ is the number of turns in the long solenoid, A is the cross-sectional area, and L is the length of the long solenoid.

Slide 4:

Now, let’s move on to the short solenoid. The self-inductance of the short solenoid is given by the formula: L2 = μ₀N₂²A’/L'
Where N₂ is the number of turns in the short solenoid, A’ is the cross-sectional area of the short solenoid, and L’ is the length of the short solenoid.

Slide 5:

The mutual inductance between the two solenoids can be calculated using the formula: M = μ₀N₁N₂A’/(L’ + L)
Where N₁ and N₂ are the number of turns in the long and short solenoids, A’ is the cross-sectional area of the short solenoid, L’ is the length of the short solenoid, and L is the length of the long solenoid.

Slide 6:

Now, let’s consider an example. We have a long solenoid with 500 turns per meter, a radius of 0.02 meters, and a length of 0.1 meters. The short solenoid has 100 turns per meter, a radius of 0.01 meters, and a length of 0.02 meters.

Slide 7:

Using the given values, we can find the self-inductance of the long solenoid using the formula: L1 = μ₀N₁²A/L
Plugging in the values, we get: L₁ = (4π × 10⁻⁷ T·m/A)(500/m)²(π(0.02m)²)/(0.1m)

Slide 8:

Simplifying the equation, we find: L₁ = 5.0 × 10⁻³ H

Slide 9:

Now, let’s find the self-inductance of the short solenoid using the formula: L₂ = μ₀N₂²A’/L'
Plugging in the values, we get: L₂ = (4π × 10⁻⁷ T·m/A)(100/m)²(π(0.01m)²)/(0.02m)

Slide 10:

Simplifying the equation, we find: L₂ = 5.0 × 10⁻⁵ H

Slide 11:

Now, let’s calculate the mutual inductance between the long and short solenoids using the formula: M = μ₀N₁N₂A’/(L’ + L)
Plugging in the values, we get: M = (4π × 10⁻⁷ T·m/A)(500/m)(100/m)(π(0.01m)²)/(0.02m + 0.1m)

Slide 12:

Simplifying the equation, we find: M = 9.0 × 10⁻⁶ H
This means that there is a mutual inductance of 9.0 × 10⁻⁶ Henrys between the long and short solenoids.

Slide 13:

The mutual inductance between two solenoids depends on the number of turns, the dimensions of the solenoids, and the separation between them.
Mutual inductance is a measure of how well the magnetic field produced by one solenoid induces a current in the other solenoid.
It plays an important role in transformers and other electromagnetic devices.

Slide 14:

Self-inductance, on the other hand, is a measure of how well a solenoid induces an electromotive force (EMF) in itself when the current passing through it changes.
It depends on the number of turns, the dimensions of the solenoid, and the material in the core.
Self-inductance opposes any change in the current passing through the solenoid.

Slide 15:

The units for self-inductance and mutual inductance are Henries (H), named after Joseph Henry, who made significant contributions to the study of electromagnetism.
1 Henry is defined as the self-inductance of a circuit that induces an EMF of 1 volt when the current through it changes at a rate of 1 ampere per second.
Inductances are usually in the range of microhenries (μH) to millihenries (mH) for most practical applications.

Slide 16:

Inductors are passive components used in electrical circuits to store and release energy in the form of a magnetic field.
They are commonly made by winding a wire into a coil, which enhances the inductance by increasing the number of turns.
Inductors are widely used in filtering circuits, oscillators, and power supplies, among other applications.

Slide 17:

The energy stored in an inductor can be calculated using the formula: E = (1/2)Li²
Where E represents the energy stored in the inductor, L is the inductance in Henries, and i is the current passing through the inductor.
This equation shows that the energy stored in an inductor is proportional to the square of the current passing through it.

Slide 18:

When the current through an inductor changes, the energy stored in the magnetic field changes as well.
This energy can be released back into the circuit when the current decreases, causing a voltage spike known as back electromotive force (EMF).
Back EMF can be a nuisance in certain applications but can also be harnessed to provide protection to circuits during transient events.

Slide 19:

In summary, we have discussed Faraday’s Law of Induction, mutual inductance, and self-inductance.
Mutual inductance measures the magnetic interaction between two coils or solenoids.
Self-inductance measures the ability of a solenoid to induce an EMF in itself when the current changes.
Inductors are used in various electrical circuits to store and release energy in the form of a magnetic field.

Slide 20:

Understanding the concepts of mutual and self-inductance is crucial in many areas of physics and engineering, such as electromagnetism, power systems, and electronic circuit design.
These concepts form the foundation for more advanced topics like transformers, inductors in AC circuits, and resonance phenomena.
It is important to grasp these concepts thoroughly to excel in the study and application of electromagnetism.

Slide 21:

Mutual inductance is used in the design and operation of transformers, electromechanical devices, and wireless power transfer systems.
Transformers are based on mutual inductance, where the changing magnetic field in one coil induces a current in the other coil.
Mutual inductance is also used in wireless charging systems, such as those used for smartphones and electric vehicles.

Slide 22:

Self-inductance plays a role in many electrical and electronic devices.
Inductors are commonly used in filter circuits to separate alternating current (AC) from direct current (DC).
They can also be used in oscillators to control the resonant frequency and maintain stability.
Inductors are critical components in power supply circuits, smoothing out fluctuations in current and voltage.

Slide 23:

The relative permeability of the core material used in the solenoids affects the inductance.
Materials with high permeability, such as iron or ferrite, enhance the inductance by increasing the strength of the magnetic field.
The core material also affects the frequency response and power handling capabilities of the inductor.

Slide 24:

The more turns a solenoid has, the greater the inductance.
Increasing the number of turns increases the magnetic flux and therefore the induced voltage.
This can be advantageous when a high inductance value is required.

Slide 25:

Inductance can vary with frequency due to the skin effect and proximity effect.
At high frequencies, the skin effect causes the current to concentrate near the surface of the wire, reducing the effective cross-sectional area and increasing the resistance.
The proximity effect occurs when multiple conductors are close together, causing interference and altering the effective inductance.

Slide 26:

The inductance of a coil can be adjusted by varying its physical dimensions or adding a ferromagnetic core material.
Increasing the length of the coil decreases the inductance, while increasing the cross-sectional area increases the inductance.
Adding a ferromagnetic core material further enhances the inductance.

Slide 27:

Mutual inductance can be utilized in various applications, such as electromagnetic sensors, voltage regulation, and wireless power transfer.
It enables devices like current transformers to measure high currents.
Mutual inductance also allows for efficient power transfer in wireless charging systems, eliminating the need for direct electrical contact.

Slide 28:

Self-inductance opposes any change in current by generating a back EMF.
This property is crucial in protecting circuits from voltage spikes and transient events.
Inductors are often used in conjunction with diodes to create voltage clamps and surge protection circuits.

Slide 29:

Mutual inductance and self-inductance are often represented by the symbols M and L, respectively, in circuit diagrams.
These symbols help to identify and understand the role of inductors in electrical circuits.
Inductance values are commonly indicated in microhenries (µH) or millihenries (mH), depending on the application.

Slide 30:

In conclusion, mutual and self-inductance are key concepts in the study of electromagnetism and electrical circuits.
Understanding these concepts is fundamental for analyzing and designing various devices, including transformers, inductors, and wireless power systems.
The ability to calculate and manipulate inductance is essential for problem-solving in physics and engineering.