Question

In a coaxial cable, the solid core carries a current \(i\) The sheath also carries a current \(i\) but in the opposite direction and has an inner radius \(a\) and an outer radius b. The current density is equally distributed over each conductor. Find an expression for the magnetic field at a distance \(a

Short answer

Answer: The magnetic field at a distance \(a < r < b\) from the center of the core is given by: \(B = \frac{\mu_0 i}{2\pi (b^2 - a^2)} * (1 - \frac{a^2}{r^2})\).

Step-by-step solution

Find the current passing through a loop between a and b

The first thing we need to do is to find the current passing through a loop between \(a\) and \(b\). We know that the current density \(J\) is equally distributed over each conductor. Thus, we can find the current passing through a loop of radius \(r\) in the sheath by integrating the current density over the area of the loop: \(I_r = \int J dA\) Since the current density is equally distributed, we know that \(J = \frac{i}{\pi a^2}\) for the core and \(J = \frac{i}{\pi (b^2 - a^2)}\) for the sheath. For a loop with an infinitesimally small thickness \(dr\) and radius \(r\) in the sheath, we have \(dA = 2 \pi r dr\). Hence, \(I_r = 2\pi r dr * \frac{i}{\pi (b^2 - a^2)}\). Integrating over the radius of the sheath, we get: \(I_r = \int_{a}^{r} 2\pi r' dr' * \frac{i}{\pi (b^2 - a^2)}\) \(I_r = \frac{2 \pi i}{\pi (b^2 - a^2)} * \int_{a}^{r} r' dr'\)

Calculate the integral to find the current

Now we will calculate the integral to find the total current within the loop: \(I_r = \frac{2 \pi i}{\pi (b^2 - a^2)} * \int_{a}^{r} r' dr'\) \(I_r = \frac{2 \pi i}{\pi (b^2 - a^2)} * \left[\frac{1}{2}r'^2\right]_{a}^{r}\) \(I_r = \frac{2 \pi i}{\pi (b^2 - a^2)} * \left[\frac{1}{2}(r^2 - a^2)\right]\) \(I_r = \frac{i}{b^2 - a^2} * (r^2 - a^2)\)

Apply Ampère's law to find the magnetic field

Now that we have the current inside the loop, we can use Ampère's law to find the magnetic field \(B\). Ampère's law states that the line integral of the magnetic field around a closed loop is equal to \(\mu_0\) times the total current passing through the loop: \(B * 2\pi r = \mu_0 I_r\)

Solve for the magnetic field B

Finally, we can solve for the magnetic field \(B\): \(B * 2\pi r = \mu_0 * \frac{i}{b^2 - a^2} * (r^2 - a^2)\) \(B = \frac{\mu_0 i}{2\pi (b^2 - a^2)} * (1 - \frac{a^2}{r^2})\) So, the magnetic field at a distance \(a < r < b\) from the center of the core is given by: \(B = \frac{\mu_0 i}{2\pi (b^2 - a^2)} * (1 - \frac{a^2}{r^2})\)

Key concepts

Ampère's Law

Ampère's Law is a fundamental principle in electromagnetism, crucial for understanding the magnetic field in coaxial cables. It relates the integrated magnetic field around a closed loop to the electric current passing through the loop. Mathematically, it is expressed as:\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \]where \( \oint \mathbf{B} \cdot d\mathbf{l} \) is the line integral of the magnetic field \( \mathbf{B} \) around a closed path, \( \mu_0 \) is the permeability of free space, and \( I_{enc} \) is the enclosed current. In essence, Ampère's Law explains how current generates a magnetic field, and it can help us find the magnitude of the magnetic field generated by a current-carrying conductor.

Current Density

Current density is a measure of how much electric current flows through a specific area in a conductor. It is defined as the current \( i \) per unit area \( A \), represented by \( J \). For a uniform distribution of current through the cross-sectional area of a conductor:\[ J = \frac{i}{A} \]In a coaxial cable, current density explains how current is spread across the surface of the core and sheath. The exercise describes the current density as equally distributed over the core and sheath, which simplifies calculations. Knowing the current density is essential for calculating how much current passes through a section of the cable.

Coaxial Cable

A coaxial cable consists of two conductive layers: a central core and an outer sheath, separated by an insulating material. These layers have distinct dimensions denoted by their inner and outer radii. In the exercise, the core and sheath carry equal but oppositely directed currents, creating an electromagnetic field. Such a cable is designed to minimize mutual interference, making it ideal for transmitting signals. Understanding the structure of a coaxial cable helps in calculating magnetic fields, as it determines how the current is distributed and interacts with the magnetic field, especially when using Ampère's Law.

Magnetic Field Calculation

To calculate the magnetic field in a coaxial cable, one needs to consider the geometry and current flow in the system. Using the steps from Ampère's Law, we first calculate the current passing through a specific path. With this current, we apply Ampère's Law to find the magnetic field along that path:\[ B \cdot 2\pi r = \mu_0 I_r \]Solving for \( B \), we find:\[ B = \frac{\mu_0 i}{2\pi (b^2 - a^2)} \times \left(1 - \frac{a^2}{r^2}\right) \]This expression gives the magnetic field at a distance \( a < r < b \) from the cable's center, showing how it varies with distance from the core. This approach exemplifies how electromagnetic theory is applied in real-world systems like coaxial cables.