Question

Infinitely long filamentary conductors are located in the \(y=0\) plane at \(x=n\) meters where \(n=0, \pm 1, \pm 2, \ldots\) Each carries \(1 \mathrm{~A}\) in the \(\mathbf{a}_{z}\) direction. (a) Find \(\mathbf{H}\) on the \(y\) axis. As a help, $$\sum_{n=1}^{\infty} \frac{y}{y^{2}+n^{2}}=\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}$$ (b) Compare your result of part \((a)\) to that obtained if the filaments are replaced by a current sheet in the \(y=0\) plane that carries surface current density \(\mathbf{K}=1 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}\).

Short answer

Question: Compare the magnetic field H generated by a series of infinitely long filamentary conductors located at x = n meters on the y=0 plane, each carrying 1A current in the az direction, with the magnetic field H generated by a current sheet carrying a surface current density K = 1 A/m. Answer: The magnetic field H generated by the filamentary conductors depends on the position y on the Y-axis, while the magnetic field H generated by the current sheet is constant, equal to 1/2 A/m. For large y values, the magnetic field generated by the filamentary conductors will be similar to that generated by the current sheet, but with a variation based on the position y on the Y-axis.

Step-by-step solution

Calculate H from the filamentary conductors (part a)

To calculate \(\mathbf{H}\) on the Y-axis due to the filaments carrying current, we first need to consider the contribution from each filament. The magnetic field \(\mathbf{H}_{n}\) due to filament n is given by the Biot-Savart Law: $$\mathbf{H}_{n}=\frac{I}{4 \pi}\oint \frac{d\boldsymbol{\ell} \times \mathbf{a}_{R}}{R^{2}}$$ Where \(R\) is the distance from the filament, \(\mathbf{a}_{R}\) is the unit vector pointing from the filament to the point on Y-axis, and d\(\boldsymbol{\ell}\) is an infinitesimal element along the filament. Since we are only interested in the \(\mathbf{H}\) components in the Y-axis, we have: $$H_{n}=\frac{1}{4 \pi}\oint \frac{(\mathbf{a}_{z}\times \mathbf{a}_{R})\cdot \mathbf{a}_{y}}{R^2} dy$$ To calculate the sum of all components, we can use the given series: $$\sum_{n=1}^{\infty} \frac{y}{y^{2}+n^{2}}=\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}$$ Thus, the magnetic field H on the Y-axis due to all filaments is given by: $$H_{y} = \frac{1}{4 \pi} \left(\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}\right)$$

Calculate H generated by the current sheet (part b)

Now we will calculate the magnetic field H on the Y-axis generated by a current sheet carrying a surface current density K = 1 A/m. As the current sheet is located on the y=0 plane, the magnetic field H will be uniformly distributed on the Y-axis. Therefore: $$H_{sheet} = \frac{K}{2} = \frac{1}{2} \ A/m$$

Compare the results obtained in parts a and b

Now we will compare the magnetic field H generated by the filamentary conductors and the current sheet. Magnetic field H due to filamentary conductors (part a): $$H_{y} = \frac{1}{4 \pi} \left(\frac{\pi}{2}-\frac{1}{2 y}+\frac{\pi}{e^{2 \pi y}-1}\right)$$ Magnetic field H due to the current sheet (part b): $$H_{sheet} = \frac{1}{2} \ A/m$$ As we can see, the magnetic field H generated by the infinitely long filamentary conductors depends on the position y on the Y-axis while the magnetic field H generated by the current sheet is constant. Moreover, for large y, the main difference between H and H_sheet is the last term, which depends on \(y\) and decreases exponentially. Overall, the magnetic field generated by the filamentary conductors will be similar to that generated by the current sheet, but with a variation based on the position y on the Y-axis.

Key concepts

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is typically represented by the symbol \(\mathbf{H}\) and is a vector field, meaning it has both a magnitude and direction. In our scenario, infinitely long conductors create magnetic fields that can be calculated using specific formulas. Understanding how these fields interact is crucial when dealing with multiple current-carrying wires.
When analyzing magnetic fields, the direction of the field lines and their density (strength) are essential. Field lines form closed loops, moving from the north pole to the south pole outside the magnet, and back within it. The closer they are, the stronger the field.
As a fundamental concept in electromagnetics, understanding magnetic fields helps in designing electric motors, transformers, and various electromagnetic devices.

Biot-Savart Law

The Biot-Savart Law is an essential equation in electromagnetics used to calculate the magnetic field generated by a steady current. It expresses the magnetic field \(\mathbf{H}\) in terms of the current distribution. For an infinitesimal length of conductor carrying current \(I\), the law is given by:

• \(\mathbf{H}_{n}=\frac{I}{4 \pi}\oint \frac{d\boldsymbol{\ell} \times \mathbf{a}_{R}}{R^{2}}\)

Here, \(d\boldsymbol{\ell}\) is the small length segment of the wire, \(\mathbf{a}_{R}\) is a vector from the element to the observation point, and \(R\) is the distance to that point.
This vector cross product gives directionality to the magnetic field. The Biot-Savart Law is particularly useful in calculating fields for currents with complex shapes and distributions.
By using this law, we can understand how each element of current contributes to the overall magnetic field, enabling more sophisticated design and analysis of electromagnetic systems.

Current Distribution

Current distribution refers to the manner in which electrical current flows through a conductor. In our problem, we are dealing with an array of infinite conductors, each with a specific distribution along the \(x\)-axis. The current flows in the \(\mathbf{a}_{z}\) direction for each conductor.
When assessing the magnetic field resulting from such distributions, it's important to consider that each conductor contributes individually to the total magnetic field. This requires integrating these contributions across the entire distribution.
A good grasp of current distribution helps in predicting how circuits behave and in designing wiring for less resistance and efficient flow. It's crucial in both simple and complex systems, from household wiring to massive power plants.

Infinitely Long Conductors

Infinitely long conductors, as the name suggests, are theoretical conductors of infinite length. They are a common simplifying assumption in physics to help make complex calculations tractable. When electricity flows through them, they generate a magnetic field around them in concentric circles.
This simplification allows us to ignore end effects and focus on the field generated around the body of the conductor. In this exercise, the conductors lie in the \(y=0\) plane, positioned along the \(x\)-axis. Each conductor carries a consistent current of \(1 \mathrm{~A}\), creating predictable and uniform magnetic fields.
Understanding these fields helps in analyzing more complex structures and leads to more accurate and simpler design calculations in real-world applications, like cables and transmission lines.