Derive the expression for the self inductance of a long solenoid of cross sectional area A and length /, having n turns per unit length ?

To derive the expression for the self-inductance of a long solenoid, we need to understand how inductance relates to the magnetic field generated by the solenoid and how this field interacts with the current flowing through it. Let's break this down step-by-step.

Understanding the Components

A solenoid is essentially a long coil of wire that generates a magnetic field when an electric current passes through it. The key parameters involved are:

• A: Cross-sectional area of the solenoid
• l: Length of the solenoid
• n: Number of turns per unit length (turns/meter)
• I: Current flowing through the solenoid
• B: Magnetic field inside the solenoid

Magnetic Field Inside the Solenoid

For a long solenoid, the magnetic field B inside is given by the formula:

B = μ₀nI

Here, μ₀ is the permeability of free space (a constant), and n is the number of turns per unit length. This formula indicates that the magnetic field is directly proportional to both the current and the number of turns.

Magnetic Flux Through the Solenoid

The total magnetic flux Φ linked with the solenoid can be expressed as:

Φ = B × A

Substituting the expression for B gives us:

Φ = (μ₀nI) × A

Inductance and Its Definition

The self-inductance L of the solenoid is defined by the relationship between the magnetic flux Φ and the current I:

L = Φ/I

Now substituting the expression for Φ into this equation, we get:

L = (μ₀nI × A) / I

Simplifying this, we find:

L = μ₀nA

Final Expression for Self-Inductance

To express n in terms of total number of turns N and length l of the solenoid, we know that:

n = N/l

Substituting this back into our equation for L, we arrive at the final expression:

L = μ₀(N/l)A

Thus, the self-inductance of a long solenoid can be summarized as:

L = (μ₀NA) / l

Conclusion

This expression gives us the self-inductance in terms of the physical dimensions of the solenoid and the properties of the material inside it. The inductance increases with more turns, a larger cross-sectional area, or a longer solenoid, providing insight into how we can design solenoids for specific applications in electromagnetism.