The critical field in a niobium-titanium superconductor is \(15 \mathrm{T}\) What current in a 5000 -turn solenoid \(75 \mathrm{cm}\) long will produce a field of this strength?

A solenoid is a type of electromagnet whose purpose is to generate a controlled magnetic field. If the solenoid is long compared to its diameter, we can make some simplifying assumptions that lead to a very useful expression for the magnetic field inside the solenoid. The magnetic field, represented by the symbol \( B \), is uniform and parallel to the axis of the solenoid.

The magnetic field inside a solenoid is directly proportional to the current \( I \) flowing through it and the number of turns per unit length \( n \), and it's given by the formula \( B = \mu_0 n I \), where \( \mu_0 \) is the permeability of free space. This relationship indicates that the field strength inside a solenoid can be adjusted by changing the current or altering the number of turns per unit length.

Magnetic Field Direction

The direction of the magnetic field can be determined by the right-hand rule. If you wrap your right hand around the solenoid with your fingers in the direction of the current, the thumb points in the direction of the magnetic field within the solenoid.

The permeability of free space, symbolized by \( \mu_0 \), is a fundamental physical constant which is the measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum. Its value is \( 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A} \) (Tesla meters per Ampere).

From Maxwell's equations, it is understood that \( \mu_0 \) is related to the speed of light in a vacuum, and it plays a critical role in the calculations of electromagnetic fields in various materials and mediums. It is particularly important in solving problems involving the magnetic field of inductors and solenoids, such as the one in our exercise.

To calculate the current required to produce a specific magnetic field in a solenoid, we need to rearrange the relationship \( B = \mu_0 n I \). Solving for the current \( I \) gives us the formula \( I = \frac{B}{\mu_0 n} \).

By substituting the desired magnetic field strength, the permeability of free space, and the number of turns per unit length into the formula, we can find the current that must flow through the solenoid to achieve this field. This calculation is vital in applications such as designing electromagnets for MRI machines or in particle accelerators where precise control of the magnetic field is essential.

The number of turns per unit length in a solenoid, often denoted as \( n \) in equations, is a measure of how many coils of wire there are in a given length of a solenoid. This is crucial because it affects the strength of the magnetic field generated by the solenoid.

To calculate \( n \), we divide the total number of turns \( N \) by the length of the solenoid \( l \): \( n = \frac{N}{l} \). A higher number of turns within a certain length results in a stronger magnetic field, assuming the current remains constant. In our exercise, the solenoid needed to have a sufficient number of coils in the specified length to achieve the impressive magnetic field of 15 Tesla, as used for a niobium-titanium superconductor.