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.