Question

Two co-axial solenoids are made by a pipe of cross sectional area \(10 \mathrm{~cm}^{2}\) and length \(20 \mathrm{~cm}\) If one of the solenoid has 300 turns and the other 400 turns, their mutual inductance is...... (a) \(4.8 \pi \times 10^{-4} \mathrm{H}\) (b) \(4.8 \pi \times 10^{-5} \mathrm{H}\) (c) \(2.4 \pi \times 10^{-4} \mathrm{H}\) (d) \(4.8 \pi \times 10^{4} \mathrm{H}\)

Short answer

The mutual inductance between the two co-axial solenoids is (b) \(4.8π × 10^{-5} \, \mathrm{H}\).

Step-by-step solution

Calculate the self-inductance of each solenoid

First, we need to find the self-inductance L1 and L2 of the solenoids using the formula: \[L = \frac{μ₀ * A * n^2}{l}\] where μ₀ = permeability of free space = \(4π × 10^{-7} \, \mathrm{T \, m/A}\), A is the cross-sectional area of the pipe, n is the number of turns per unit length and l is the length of the solenoid. For solenoid 1 \(n_1 = \frac{300}{0.2} \, \mathrm{turns/m}\) For solenoid 2 \(n_2 = \frac{400}{0.2} \, \mathrm{turns/m}\)

Calculate the mutual inductance between the solenoids

To find the mutual inductance M, we use the following formula: \[M = \frac{μ₀ * A * n_1 * n_2}{l}\] Insert the values and perform the calculation: \(M = \frac{4π × 10^{-7} * 10^{-2} * (\frac{300}{0.2}) * (\frac{400}{0.2})}{0.2}\)

Calculate the answer

Now, simplify and solve the expression for M: \(M = 4.8π × 10^{-5} \, \mathrm{H}\) So the answer is (b) \(4.8π × 10^{-5} \, \mathrm{H}\).

Key concepts

Self-Inductance

Self-inductance is a property of a solenoid that allows it to induce an opposing voltage when the current flowing through it changes. This is expressed through its property called inductance. When you have a solenoid with self-inductance, it can be visualized as a coil of wire that creates a magnetic flux in its interior when an electric current flows through it.

The self-inductance of a solenoid can be calculated with the formula:

• \(L = \frac{μ₀ * A * n^2}{l}\)

where:

• \(L\) is the self-inductance
• \(μ₀\) is the permeability of free space
• \(A\) is the cross-sectional area
• \(n\) is the number of turns per unit length
• \(l\) is the length of the solenoid

This formula shows that the self-inductance depends on the physical properties of the solenoid, such as its size, the material it's made of, and its geometric arrangement. A higher number of turns or a larger cross-sectional area increases the inductance, while a longer solenoid decreases it.

Solenoids

A solenoid is a long coil of wire wrapped in the shape of a cylinder. Solenoids are widely used because they can generate a uniform magnetic field in their interior when an electric current passes through them. This makes them ideal for applications where a uniform magnetic field is needed, like in MRI machines or electrical relays.

Solenoids work on the principle of electromagnetism, where the electric current generates a magnetic field. Having tightly wound coils enhances this effect, making solenoids very efficient at creating concentrated magnetic fields. The number of turns in the solenoid and the current flowing through it are crucial factors that determine the strength of the magnetic field.

The magnetic field generated inside a solenoid is practically uniform across the cross-sectional area, which is why solenoids are often preferred in applications requiring precision. Their effectiveness is so pronounced that replacing the coil material or adjusting dimensions directly alters the magnetic properties.

Turns Per Unit Length

The term "turns per unit length" refers to how many loops or coils make up a solenoid over a specific length. It's a way to express the density of the turns along the length of the solenoid. The more densely packed the turns are, the stronger the resultant magnetic field will be for a given current.

In both the self-inductance and mutual inductance formulas, you'll see the number of turns per unit length, \(n\), as a crucial term. This is calculated by dividing the total number of turns by the length of the solenoid:

• \(n = \frac{N}{l}\)

where:

• \(N\) is the total number of turns
• \(l\) is the length of the solenoid

The greater the value of \(n\), the more concentrated the magnetic field becomes.

Overall, turns per unit length play a vital role in determining the inductance of a solenoid. Therefore, when designing solenoids for specific applications, adjustments to the turns per unit length offer a method of tuning the solenoid's performance to fit the required magnetic field strength and response.