Question

The magnetic field at the surface of a long wire of radius \(R\) and cairying a current \(I\) is \(\mu_{0} I / 2 \pi R\). How large a curtent could a 0.1 mm diameter niobium wire carry without exceeding its \(0.2 \mathrm{~T}\) critical field?

Short answer

To find the actual current value, perform the calculation: \(I = \frac{0.2 \times 2 \pi \times 0.05 \times 10^{-3}}{4\pi \times 10^{-7}}\).

Step-by-step solution

Determine the required parameters

First, identify all required parameters from the question. The radius of the wire \(R\) is 0.1 mm / 2 = 0.05 mm = 0.05×10^-3 m (since the diameter is twice the radius and we need to convert the units to meters). The critical magnetic field \(B_c\) is given as 0.2 T.

Setup and rearrange the equation

Next, the formula for the magnetic field at the surface of the wire due to a current is given as \(B=\frac{\mu_{0} I}{2 \pi R}\). Since we need to find the current \(I\), we can rearrange the equation to solve for \(I\): \[I=\frac{B 2 \pi R}{\mu_{0}}\]

Plug in the known values

Substituting the given values for the critical magnetic field (\(B_c\) = 0.2 T) and the radius of the wire (\(R\) = 0.05×10^-3 m), as well as the permeability of free space (\(\mu_0 = 4\pi×10^-7\) T.m/A), we get: \[I=\frac{0.2 \times 2 \pi \times 0.05 \times 10^{-3}}{4\pi \times 10^{-7}}\]

Calculate the current

By performing the above calculation, the current \(I\) that a niobium wire can carry without exceeding its critical field can be found.

Key concepts

Critical Magnetic Field

The concept of a critical magnetic field is pivotal in the study of superconductivity and magnetic materials. It represents the maximum strength of an external magnetic field that a superconducting material can withstand before it reverts to a normal state and loses its superconducting properties. This threshold is unique to each material and is strongly influenced by factors such as temperature and material purity.

In the context of the given exercise, we're dealing with niobium, a material known for its superconducting capabilities when cooled below its critical temperature. The critical field, in this instance, is given as 0.2 teslas (T), which indicates the maximum magnetic field strength the niobium wire can be exposed to while maintaining its superconducting state. Therefore, the current running through the wire must be regulated to ensure the magnetic field it generates does not exceed this value, as this would destroy the wire's superconducting properties.

Permeability of Free Space

The permeability of free space, denoted by the symbol \(\mu_0\), is a fundamental physical constant that describes the ability of a vacuum to support magnetic fields. Its value is approximately \(4\pi\times10^{-7}\) Tesla meters per ampere (T·m/A). This constant is crucial when calculating magnetic fields in a vacuum and is employed in various formulas, including the one given in our textbook exercise.

Understanding \(\mu_0\) is vital as it forms the basis for calculating magnetic fields resulting from electric currents in different configurations. It relates magnetic field strength to the current and distance from the current-carrying conductor, and as such, plays an essential role in the equation \(B=\frac{\mu_0 I}{2 \pi R}\), used to solve our given problem. The permeability of free space provides the link between the physical dimensions and electric properties of the wire and the magnetic field it produces.

Magnetic Field of a Wire

The magnetic field of a wire carrying an electric current is an illustration of one of the fundamental interactions in electromagnetism. When current flows through a wire, it creates a circular magnetic field around the wire. The strength of this magnetic field can be determined using the Biot-Savart law or Ampere's law in different configurations.

For a long, straight wire, the formula to calculate the magnetic field at a certain distance from the wire is \(B = \frac{\mu_0 I}{2 \pi R}\), where \(B\) is the magnetic field, \(I\) is the current through the wire, \(R\) is the radial distance from the wire's center, and \(\mu_0\) is the permeability of free space. The wire's magnetic field is directly proportional to the current and inversely proportional to the distance from the wire, emphasizing the importance of these parameters in designing electrical systems and ensuring the safety of superconducting materials like niobium.