Question

An ideal solenoid has length \(\ell\). If the windings are compressed so that the length of the solenoid is reduced to \(0.50 \ell,\) what happens to the inductance of the solenoid?

Short answer

The inductance of the solenoid doubles when its length is halved.

Step-by-step solution

Understanding Solenoid Inductance

The inductance of an ideal solenoid is given by the formula \(L = \frac{\mu_0 N^2 A}{\ell}\), where \(\mu_0\) is the permeability of free space, \(N\) is the number of turns, \(A\) is the cross-sectional area, and \(\ell\) is the length of the solenoid.

Identify the Change in Length

We need to calculate how the inductance \(L\) of the solenoid changes when its length \(\ell\) is reduced to \(0.5\ell\). This change implies that the new length \(\ell' = 0.5\ell\).

Substitute and Simplify

When the length of the solenoid is reduced, the expression for inductance \(L'\) becomes \(L' = \frac{\mu_0 N^2 A}{0.5\ell}\). Simplifying this expression, the inductance changes to \(L' = \frac{\mu_0 N^2 A}{\ell}\cdot 2 = 2L\).

Conclude the Inductance Change

From the calculation, the inductance of the solenoid doubles when its length is halved. Thus, the new inductance is \(2L\) where \(L\) is the original inductance.

Key concepts

Ideal Solenoid

An ideal solenoid is a type of electromagnetic device used in various applications ranging from motors to sensors. It consists of a coil of wire, typically with a large number of turns, wound tightly in a helical form. An ideal solenoid is an abstraction, meaning it assumes perfect conditions in theoretical physics.

In an ideal solenoid:

• The turns are closely packed together without any gaps between them.
• The length of the solenoid is significantly greater than its diameter, ensuring the magnetic field inside is uniform and predominantly along the axis of the solenoid.
• There is negligible resistance or energy loss along the wire.

By simplifying these conditions, calculations regarding electromagnetic fields and inductance become more straightforward, making it easier to predict the solenoid's behavior under different conditions.

Inductance Formula

The formula to calculate the inductance of a solenoid, which measures its ability to store energy in a magnetic field, is given by:

• \[ L = \frac{\mu_0 N^2 A}{\ell} \]

In this formula:

• \(L\) represents the inductance.
• \(\mu_0\) is the permeability of free space, a constant with a value of approximately \(4\pi \times 10^{-7}\, \text{T}\cdot\text{m/A}\).
• \(N\) is the number of turns of the wire in the solenoid.
• \(A\) is the cross-sectional area of the solenoid, which affects the amount of magnetic flux it can contain.
• \(\ell\) is the length of the solenoid.

Each of these factors plays a crucial role in determining the inductance. For instance, a greater number of turns \(N\) or larger cross-sectional area \(A\) results in higher inductance. Similarly, decreasing the length \(\ell\) of the solenoid increases the inductance, assuming other parameters are constant.

Permeability of Free Space

The permeability of free space, denoted as \(\mu_0\), is a fundamental constant in physics that describes the extent to which a magnetic field can permeate a vacuum. Its value is approximately \(4\pi \times 10^{-7}\, \text{T}\cdot\text{m/A}\) (Tesla meter per Ampere). This constant appears in many electromagnetic formulas as it relates to the magnetic fields found in free space.

The role of \(\mu_0\):

• It is crucial in the calculation of the inductance of a solenoid, as it directly affects how easily the magnetic field lines circulate within and around the solenoid.
• The constant helps in defining the strength of the magnetic field generated by a current flowing through the solenoid.
• It serves as a bridge between electricity and magnetism, reflecting how one can influence the other.

Understanding \(\mu_0\) is essential when studying configurations such as solenoids, where the relationship between electricity and magnetism is a key consideration.

Inductance Change with Length

One of the significant considerations in the design and operation of a solenoid is understanding how its inductance changes with its length. As seen from the formula \(L = \frac{\mu_0 N^2 A}{\ell}\), the inductance is inversely proportional to the length \(\ell\), implying that as the length decreases, the inductance increases proportionally.

Effects of changing length:

• When the length of a solenoid is reduced to half, as in this exercise, the inductance doubles, meaning \(L' = 2L\).
• This change occurs because the decreased length results in a more concentrated magnetic field.
• Such changes are instrumental in tweaking the performance of devices relying on solenoids, especially when space or design constraints limit changes to other parameters such as the number of turns or the cross-sectional area.

This relationship highlights the flexibility in solenoid design, wherein altering the physical dimensions can tailor the inductance to suit specific needs in electronic circuits and electromagnetic systems.