Question

A toroid having a cross section of rectangular shape is defined by the following surfaces: the cylinders \(\rho=2\) and \(\rho=3 \mathrm{~cm}\), and the planes \(z=1\) and \(z=2.5 \mathrm{~cm}\). The toroid carries a surface current density of \(-50 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m}\) on the surface \(\rho=3 \mathrm{~cm}\). Find \(\mathbf{H}\) at the point \(P(\rho, \phi, z):(a) P_{A}(1.5 \mathrm{~cm}, 0\), \(2 \mathrm{~cm}) ;\left(\right.\) b) \(P_{B}(2.1 \mathrm{~cm}, 0,2 \mathrm{~cm}) ;\) (c) \(P_{C}(2.7 \mathrm{~cm}, \pi / 2,2 \mathrm{~cm}) ;\) (d) \(P_{D}(3.5 \mathrm{~cm},\), \(\pi / 2,2 \mathrm{~cm})\)

Short answer

Question: Determine the magnetic field intensity (H) at points PA(1.5 cm, 0, 2 cm), PB(2.1 cm, 0, 2 cm), PC(2.7 cm, π/2, 2 cm), and PD(3.5 cm, π/2, 2 cm) for a toroid with a surface current density of -50 𝑎𝑧 A/m on the surface ρ=3 cm. Answer: The magnetic field intensity (H) at the four points are as follows: - H(PA) = 0 - H(PB) = -47.62 𝑎𝜑 A/m - H(PC) = -55.56 𝑎𝜑 A/m - H(PD) = 0

Step-by-step solution

1. Identify given parameters and the Ampere's Circuital Law formula

The given parameters are: - The toroid has a cross section of rectangular shape and is defined by the cylindrical surfaces at ρ=2 cm and ρ=3 cm, and the plane surfaces at z=1 cm and z=2.5 cm. - The toroid has a surface current density of -50 𝑎𝑧 A/m on the surface ρ=3 cm. - We need to find the magnetic field intensity at the four points: 𝑃𝐴(1.5 cm, 0, 2 cm), 𝑃𝐵(2.1 cm, 0, 2 cm), 𝑃𝐶(2.7 cm, π/2, 2 cm), and 𝑃𝐷(3.5 cm, π/2, 2 cm). According to Ampere's Circuital Law, the line integral of the magnetic field intensity (H) over a closed path, denoted as ∮𝐻·d𝐿, is equal to the net current enclosed by the path (I), which can be represented as: ∮𝐻·d𝐿 = 𝐼

2. Calculate the magnetic field intensity (H) at point PA

For point PA, we can see that it lies outside the toroid since ρ<2 cm. Therefore, there is no current enclosed by the closed path. By applying Ampere's Law: ∮𝐻·d𝐿 = 0 Thus, H(PA) = 0

3. Calculate the magnetic field intensity (H) at point PB

For point PB, we can see that it lies inside the toroid with ρ>2 cm and ρ<3 cm. Therefore, we only need to consider the current within the closed path up to ρ=2 cm, since it is the nearest surface with a current. Due to the azimuthal symmetry of the toroid, the magnetic field intensity H can be assumed to have only an ϕ component (𝐻ϕ): ∮𝐻ϕd𝐿 = -50 A/m × (2π × 2 cm) Hence, Hϕ = (-50 A/m) × (2π × 2 cm) / (2π × 2.1 cm) = -47.62 𝑎𝜑 A/m Thus, H(PB) = -47.62 𝑎𝜑 A/m

4. Calculate the magnetic field intensity (H) at point PC

For point PC, it lies on the surface where the toroid has a surface current density, ρ=3 cm. So, we need to consider the current enclosed by the entire cross-sectional area of the toroid: ∮𝐻ϕd𝐿 = -50 A/m × (2π × 3 cm) Since the path goes through point PC, it is equal to Hϕ × (2π × 2.7 cm): Hϕ = (-50 A/m) × (2π × 3 cm) / (2π × 2.7 cm) = -55.56 𝑎𝜑 A/m Thus, H(PC) = -55.56 𝑎𝜑 A/m

5. Calculate the magnetic field intensity (H) at point PD

For point PD, it lies outside the toroid since ρ>3 cm. Therefore, there is no net enclosed current in the closed path. By applying Ampere's Law: ∮𝐻·d𝐿 = 0 Thus, H(PD) = 0 Now, we have the magnetic field intensity at all four points: - H(PA) = 0 - H(PB) = -47.62 𝑎𝜑 A/m - H(PC) = -55.56 𝑎𝜑 A/m - H(PD) = 0

Key concepts

Toroid

A toroid is a fascinating shape often used in physics and engineering, especially in studies involving magnetic fields. Imagine a doughnut-like figure, but instead of delicious flavors, we have wires coiled around it to create a toroid. In more technical terms, a toroid is a toroidal coil or wire wound in a doughnut shape. This configuration is particularly useful for generating magnetic fields. It confines magnetic field lines within the center of its coils, which means electromagnetic fields rarely escape outside. This property is essential for devices like transformers and inductors. In this exercise, the toroid is defined by the circular surfaces at distances

• Inner radius: \( \rho = 2 \, \mathrm{cm} \)
• Outer radius: \( \rho = 3 \, \mathrm{cm} \)

Additionally, it has flat surfaces defined by planes at:

• Lower plane: \( z = 1 \, \mathrm{cm} \)
• Upper plane: \( z = 2.5 \, \mathrm{cm} \)

These boundaries produce a clear and confined magnetic field structure within the toroidal coil.

Magnetic Field Intensity

Magnetic field intensity, represented by \( \mathbf{H} \), is a measure of the magnetizing force within a material. It plays a crucial role in understanding how magnetic fields behave, especially inside materials. In the context of a toroid, we're interested in how strong the magnetic field inside the toroid is and how it influences points inside and around it. According to Ampere's Circuital Law, the magnetic field can be determined by the current distribution in a closed loop.For instance, at points inside the toroid's windings, magnetic field intensity will be determined by integrating the current density over the loop enclosed by the chosen points. However, outside the toroid, there appears to be no net magnetic field because the fields cancel out. This is why points like \( P_D = (3.5 \, \mathrm{cm}, \, \phi, \, 2 \, \mathrm{cm}) \) show zero magnetic field intensity in this context.

Cylindrical Coordinates

Cylindrical coordinates are a way to extend two-dimensional polar coordinates to three dimensions. This system uses three parameters

• \( \rho \): the radial distance from the origin,
• \( \phi \): the angular position around the z-axis, and
• \( z \): the height above the xy-plane.

This coordinate system matches nicely with the geometry of a toroid, which often wraps around a cylinder. In this exercise, all points are defined using cylindrical coordinates, such as - \( P_A = (1.5 \, \mathrm{cm}, \, 0, \, 2 \, \mathrm{cm}) \) and - \( P_C = (2.7 \, \mathrm{cm}, \, \pi/2, 2 \, \mathrm{cm}) \).These coordinates help us easily navigate the positions of points like where the magnetic field intensity needs to be calculated. Understanding the spatial layout using these parameters allows us to determine if a point lies within or outside the toroid, affecting the net magnetic fields experienced there.

Surface Current Density

Surface current density, denoted as \( \mathbf{K} \), is a measure of the current flowing through a unit area of surface. It is essentially talking about how closely packed current carriers are on a surface. In our toroidal configuration, the surface current density is significant because it influences the magnetic field the toroid generates.For the given toroid, at \( \rho = 3 \, \mathrm{cm} \),we know the surface current density is \( -50 \, \hat{\mathbf{a}}_z \, \mathrm{A/m} \).This negative sign indicates the direction of the current, which is crucial for determining the orientation of the magnetic fields produced. The density is a crucial factor in applying Ampere’s Circuital Law, used to calculate the magnetic field intensity. When dealing with points such as \( P_B = (2.1 \, \mathrm{cm}, \, 0, \, 2 \, \mathrm{cm}) \), inside the toroid, the current distribution within the closed path contributes to calculating the resulting field magnitude and direction.