Question

What is the magnitude of the magnetic field inside a long, straight tungsten wire of circular cross section with diameter \(2.4 \mathrm{~mm}\) and carrying a current of \(3.5 \mathrm{~A}\), at a distance of \(0.60 \mathrm{~mm}\) from its central axis?

Short answer

Answer: The magnitude of the magnetic field inside the tungsten wire at a distance of 0.60 mm from its central axis is approximately 2.92 x 10^-3 T.

Step-by-step solution

Write down the formula for the magnetic field inside a wire

To calculate the magnetic field at a distance r from the central axis of a long straight wire carrying a current I, we use Ampere's Law formula: \[B = \frac{\mu_{0} I r}{2 \pi R^2}\] where B is the magnetic field, \(\mu_{0}\) is the permeability of free space (\(4\pi \times 10^{-7} \mathrm{~T \cdot m / A}\)), I is the current, r is the distance from the central axis, and R is the radius of the wire. Keep in mind that the wire has a diameter of \(2.4 \mathrm{~mm}\), so the radius R is half of that: \[R = \frac{2.4 \mathrm{~mm}}{2} = 1.2 \mathrm{~mm} = 1.2 \times 10^{-3} \mathrm{~m}\]

Plug in the given values

Now, we can plug in all the given values into the equation and solve for B. The current I is \(3.5 \mathrm{~A}\), and the distance r from the central axis is \(0.60 \mathrm{~mm} = 0.60 \times 10^{-3} \mathrm{~m}\): \[B = \frac{4\pi \times 10^{-7} \mathrm{~T \cdot m / A} \cdot 3.5 \mathrm{~A} \cdot 0.60 \times 10^{-3} \mathrm{~m}}{2 \pi \cdot (1.2 \times 10^{-3} \mathrm{~m})^2}\]

Calculate the magnetic field magnitude

Now, simplify the expression and calculate the magnetic field magnitude: \[B = \frac{4\pi \times 10^{-7} \mathrm{~T \cdot m / A} \cdot 3.5 \mathrm{~A} \cdot 0.60 \times 10^{-3} \mathrm{~m}}{2 \pi \cdot 1.44 \times 10^{-6} \mathrm{~m^2}} = \frac{7 \times 10^{-7} \mathrm{~T \cdot m / A} \cdot 0.60 \times 10^{-3} \mathrm{~m}}{1.44 \times 10^{-6} \mathrm{~m^2}}\] \[B \approx 2.92 \times 10^{-3} \mathrm{~T}\] So, the magnitude of the magnetic field inside the tungsten wire at a distance of \(0.60 \mathrm{~mm}\) from its central axis is approximately \(2.92 \times 10^{-3} \mathrm{~T}\).

Key concepts

Ampere's Law

Ampere's Law is a fundamental principle used to relate the magnetic field around a closed loop to the current passing through it. Essentially, Ampere’s Law states that the magnetic field integrated along a closed path is equal to the permeability of free space times the current passing through the path. This law is mathematically expressed as:

• \( \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I \)

Here, \( \mathbf{B} \) is the magnetic field, \( d\mathbf{l} \) is an infinitesimal element of the path, \( \mu_0 \) is the permeability of free space, and \( I \) is the current enclosed by the loop.
In situations involving long, straight conductors, Ampere's Law simplifies the calculation of magnetic fields, especially for symmetrical configurations like cylindrical wires. By choosing an Amperian loop that fits the symmetry of the wire, the magnetic field can be calculated accurately.

Current in Wire

The current flowing through a conductor, such as a wire, is the rate at which electric charge passes through a cross-sectional area of the wire. It is usually denoted by \( I \) and measured in amperes (A). In this problem, we're dealing with a straight tungsten wire carrying a current of \( 3.5 \) A.
This current generates a magnetic field around the wire due to the movement of charges. Current in a wire can be visualized as a stream of electrons moving through the conductor, and the amount of current depends on two main factors:

• The number of charges passing through the wire per unit time.
• The charge each of these particles carries.

Understanding the relationship between current and magnetism is key to calculating the magnetic field within or around the wire.

Permeability of Free Space

Permeability of free space, symbolized as \( \mu_0 \), is a constant that quantifies the ability of a vacuum to sustain a magnetic field. The value of \( \mu_0 \) is approximately \( 4\pi \times 10^{-7} \) T·m/A.
This constant is vital in calculating magnetic fields using Ampere's Law. It reflects how much magnetic field is produced in a vacuum by a given current, serving as a bridge between electric currents and the magnetic fields they generate.
In essence, permeability of free space helps understand how electromagnetic fields propagate in a vacuum, playing a crucial role in many electromagnetic calculations.

Magnetic Field Calculation

To calculate the magnetic field inside a wire, we use the formula derived from Ampere's Law:

• \( B = \frac{\mu_{0} I r}{2 \pi R^2} \)

Here, \( B \) is the magnetic field, \( I \) is the current, \( r \) is the distance from the central axis, and \( R \) is the wire's radius stored as \( 1.2 \times 10^{-3} \) m for this particular problem.
This formula enables the understanding of how the magnetic field varies within the wire. The magnetic field is directly proportional to the current in the wire and the radial distance \( r \), yet inversely proportional to the square of the radius \( R \).
By substituting known values into this equation, you can find the magnetic field's strength at any point inside the wire, as demonstrated in the given problem. This step-by-step approach ensures clarity in calculation.