Question

A current filament on the \(z\) axis carries a current of \(7 \mathrm{~mA}\) in the \(\mathbf{a}_{z}\) direction, and current sheets of \(0.5 \mathrm{a}_{z} \mathrm{~A} / \mathrm{m}\) and \(-0.2 \mathrm{a}_{z} \mathrm{~A} / \mathrm{m}\) are located at \(\rho=1 \mathrm{~cm}\) and \(\rho=0.5 \mathrm{~cm}\), respectively. Calculate \(\mathbf{H}\) at: \((\) a \() \rho=0.5 \mathrm{~cm} ;(b) \rho=\) \(1.5 \mathrm{~cm} ;(c) \rho=4 \mathrm{~cm} .(d)\) What current sheet should be located at \(\rho=4 \mathrm{~cm}\) so that \(\mathbf{H}=0\) for all \(\rho>4 \mathrm{~cm}\) ?

Short answer

Question: Calculate the magnetic field intensity H at different positions (a) ρ = 0.5 cm, (b) ρ = 1.5 cm, (c) ρ = 4 cm with respect to a system of current-carrying wires and sheets. Also, find the current sheet required to cancel the magnetic field outside ρ=4cm. Answer: To find the magnetic field intensity H at each position, follow these steps: 1. Calculate the magnetic field intensity due to the current filament and the current sheets using the Biot-Savart law and Ampère's law, respectively. 2. Calculate the total magnetic field intensity H for each case (a), (b), and (c) by summing the magnetic field intensities due to the current sources. 3. To cancel the magnetic field outside ρ = 4cm, find the required current density K_req for an additional current sheet such that H_new = -H_c. Complete these calculations to find the magnetic field intensities at each position and the required current density for the additional current sheet.

Step-by-step solution

Find the magnetic field due to the current filament

Start by finding the magnetic field intensity H due to the current filament at a distance ρ from the z-axis. For a current filament, the magnetic field intensity H can be calculated using the Biot-Savart law: H_filament = \frac{I}{2πρ} \mathbf{a}_{φ}, where I = 7mA is the current flowing in the filament.

Find the total magnetic field due to the current sheets

Each current sheet creates a magnetic field, and we need to find the total magnetic field intensity due to these current sheets. From Ampère's law, the magnetic field intensity due to a current sheet at a distance ρ from the z-axis is given by: H_sheet(ρ) = \frac{K}{2} \mathbf{a}_{φ}, where K is the sum of the current densities within the given ρ.

Calculate H for each case (a), (b), and (c)

(a) ρ = 0.5 cm: Since this point is on the same location as the second current sheet, we need to consider only the current filament and the first current sheet. Calculate the magnetic field intensities due to these sources and add them up: H_a = H_filament + H_sheet(ρ = 0.5 cm) (b) ρ = 1.5 cm: This point is between the first and second current sheets. Calculate the magnetic field intensities due to both the current sheets and the current filament, and sum them up: H_b = H_filament + H_sheet(ρ = 1.5 cm) (c) ρ = 4 cm: This point is outside both current sheets, so we need to consider all the current sources. Calculate the magnetic field intensities due to both the current sheets and the current filament, and sum them up: H_c = H_filament + H_sheet(ρ = 4 cm)

Calculate the required current sheet to cancel the magnetic field

(d) To cancel the magnetic field for all ρ > 4 cm, we need to find the current sheet that opposes the existing magnetic field at ρ = 4 cm. This means that the new current sheet should have a magnetic field intensity equal and opposite to the existing magnetic field at that point. To find the required current density K_req for the current sheet at ρ = 4 cm, solve the equation: H_new = -H_c, where H_new is the magnetic field intensity due to the new current sheet, calculated using Ampère's law: H_new = \frac{K_req}{2} \mathbf{a}_{φ}. Find K_req to make H = 0 for all ρ > 4 cm.

Key concepts

Magnetic Field Intensity

The magnetic field intensity, denoted as \( \mathbf{H} \), is an important parameter in electromagnetics. It describes the strength and direction of a magnetic field generated by electric currents. Magnetic field intensity is not only dependent on the sources of the magnetic field, like current-carrying conductors, but also on the medium through which the field propagates.
One can think of \( \mathbf{H} \) in simple terms as the vector representation of magnetic force exerted within an electromagnetic field. It is crucial to calculate \( \mathbf{H} \) when analyzing the effects of multiple electromagnetic sources, such as current filaments and sheets, as shown in this exercise.

Biot-Savart Law

The Biot-Savart Law offers a fundamental approach to determining the magnetic field intensity generated by a filamentary current. In formula terms, it states:

\[ H_{\text{filament}} = \frac{I}{2\pi \rho} \mathbf{a}_{\phi} \]

where:\

• \( I \) represents the current magnitude flowing through the filament. In this problem, it is 7 mA.
• \( \rho \) denotes the perpendicular distance from the current filament's axis to the point under consideration.
• \( \mathbf{a}_{\phi} \) is the azimuthal unit vector giving direction of the magnetic field relative to cylindrical coordinates.

With Biot-Savart Law, one can systematically find \( \mathbf{H} \) at various points around a current filament, aiding in the comprehensive understanding of electromagnetic fields.

Ampère's Law

Ampère's Law enables simpler calculations of magnetic fields around currents, especially distributed current sources like sheets. In its fundamental equation, it ties the path integral of magnetic field intensity around a closed loop to the current through it.

For a current sheet, Ampère's Law simplifies the task of finding magnetic field intensity as follows:

\[ H_{\text{sheet}(\rho)} = \frac{K}{2} \mathbf{a}_{\phi} \]

where:

• \( K \) is the current density of the sheet, representing how much current flows per unit width.
• \( \mathbf{a}_{\phi} \) signifies the direction of the field.

Ampère's Law proves effective in predicting the combined effects of multiple current sources, such as filament and sheet currents, guiding precise magnetic field assessments.

Current Filament

A current filament is a simplification assuming current flows in a narrow, concentrated path aligned along a certain axis. In the context of this problem, the current filament resides on the \( z \)-axis, carrying a current of 7 mA directed along \( \mathbf{a}_{z} \).
Despite its simplicity, the current filament model is invaluable in predicting the magnetic fields created by real-world narrow conductors. It reduces complex magnetic field calculations into straightforward applications of the Biot-Savart Law. Understanding these simplifications assists in constructing more accurate models of magnetic field behaviors crucial in electromagnetic analysis.

Current Sheet

A current sheet is a conceptual model that describes a broad, continuous distribution of currents over a surface. This model becomes instrumental when currents spread over relatively large areas compared to their thickness.
The disharmony between sheets at different distances, such as \( \rho = 0.5 \text{ cm} \) and \( \rho = 1 \text{ cm} \) in this scenario, highlights the diverse contributions each current sheet can make. Given current densities of 0.5 and -0.2 A/m, they illustrate how various sheets can summon differing magnetic fields around them.
This analysis comes in handy in visualizing how current distribution in sheets impacts the resulting magnetic field and aids in solving problems like canceling field effects for regions beyond a specific point.