Question

In spherical coordinates, the surface of a solid conducting cone is described by \(\theta=\pi / 4\) and a conducting plane by \(\theta=\pi / 2 .\) Each carries a total current I. The current flows as a surface current radially inward on the plane to the vertex of the cone, and then flows radially outward throughout the cross section of the conical conductor. \((a)\) Express the surface current density as a function of \(r ;(b)\) express the volume current density inside the cone as a function of \(r ;(c)\) determine \(\mathbf{H}\) as a function of \(r\) and \(\theta\) in the region between the cone and the plane; \((d)\) determine \(\mathbf{H}\) as a function of \(r\) and \(\theta\) inside the cone.

Short answer

In this problem, we have a solid conducting cone and a conducting plane, both carrying a total current I. We found the surface current density as a function of r to be \(K_s(r) = \frac{I}{\pi r^2}\) and the volume current density inside the cone as a function of r to be \(J_v(r) = \frac{3 I}{\pi h r^2 \tan^2(\theta)}\). The magnetic field H between the cone and the plane is \(\mathbf{H}(r, \theta) = \frac{I}{2 \pi r} \hat{\phi}\) for \(\pi/4 \le \theta \le \pi/2\), and inside the cone, the magnetic field is \(\mathbf{H}(r,\theta) = \frac{I}{2 \pi r} \hat{\phi}\) for \(0 \le \theta \le \pi/4\).

Step-by-step solution

Surface current density as a function of r

Since the current I is flowing radially inward on the plane, and the total current is constant, the surface current density must decrease as we move away from the vertex of the cone. We know that the area of a circle with radius r is given by \(A = \pi r^2\). So, the surface current density \(K_s\) as a function of \(r\) can be defined as: $$K_s(r) = \frac{I}{\pi r^2}.$$

Volume current density inside the cone as a function of r

For the volume current density \(J_v\), we can consider the current flowing in the radial direction within the cross-section of the conducting cone. Since the current is flowing radially outward through the entire cross-section of the conical conductor, we need to find the ratio of the current I to the volume V(r) of the cone. \(r = R(\theta) = R \tan(\theta) => R = \frac{r}{\tan(\theta)}\) The volume of a cone is given by \(V(r) = \frac{1}{3}\pi R^2 h = \frac{1}{3}\pi (\frac{r}{\tan(\theta)})^2 h\). Thus, the volume current density \(J_v(r)\) can be defined as: $$J_v(r) = \frac{I}{V(r)} = \frac{3 I}{\pi h r^2 \tan^2(\theta)}.$$

Magnetic field H between the cone and the plane

In the region between the cone and the plane, we can use Ampere's law to find the magnetic field H. Since the current is radially inward on the plane and the magnetic field lines are concentric circles around the vertex, we can write Ampere's law as: $$\oint \mathbf{H} \cdot d\mathbf{l} = I_{enclosed}.$$ For a circle with radius r centered at the vertex of the cone, the path around the circle is \(dl = 2 \pi r\). Thus, H(r) is: $$H(r) = \frac{I_{enclosed}}{2 \pi r}.$$ To find the enclosed current, we can use the surface current density: $$I_{enclosed} = K_s(r) \pi r^2 = I.$$ So the magnetic field between the cone and the plane is: $$\mathbf{H}(r, \theta) = \frac{I}{2 \pi r} \hat{\phi}.\text{ for } \pi/4 \le \theta \le \pi/2$$

Magnetic field H inside the cone

Inside the cone, the magnetic field is generated by the volume current density \(J_v(r)\). We can again use Ampere's law to find the magnetic field H. Since the current is radially outward within the cone and the magnetic field lines are concentric circles around the vertex, we can write Ampere's law as: $$\oint \mathbf{H} \cdot d\mathbf{l} = I_{enclosed}.$$ For a circle with radius r centered at the vertex of the cone, the path around the circle is \(dl = 2 \pi r\). Thus, H(r) is: $$H(r) = \frac{I_{enclosed}}{2 \pi r}.$$ To find the enclosed current, we can use the volume current density: $$I_{enclosed} = J_v(r) V(r) = I.$$ So the magnetic field inside the cone is: $$\mathbf{H}(r,\theta) = \frac{I}{2 \pi r} \hat{\phi}.\text{ for } 0 \le \theta \le \pi/4$$

Key concepts

Spherical Coordinates

In spherical coordinates, positions are defined using three parameters: \(r\), \(\theta\), and \(\phi\). This system is particularly useful for problems with spherical symmetry. The parameter \(r\) measures the radial distance from the origin to a point in space.
The angle \(\theta\) is the polar angle, which determines the inclination from the positive z-axis. Finally, \(\phi\) is the azimuthal angle in the xy-plane from the positive x-axis.
This coordinate system is crucial when working with problems involving shapes like spheres or cones, as it simplifies the description of such surfaces.
Spherical coordinates are commonly used in electromagnetism and physics to describe the geometry of three-dimensional spaces.

Surface Current Density

Surface current density, often symbolized as \(K_s\), is a measure of current per unit length moving across a surface. It is analogous to volume current density, but instead of being distributed through a volume, it is distributed along a surface like a conducting plane or sheet.
The unit for surface current density is amperes per meter (A/m).

• In the context of the cone and plane, the surface current density describes how the current flows radially inward on the conducting plane.
• Due to the constant total current \(I\), the density decreases as the radius \(r\) increases, hence \(K_s(r) = \frac{I}{\pi r^2}\).

Understanding \(K_s\) is essential for analyzing how current is distributed over a surface and its effects on surrounding electromagnetic fields.

Volume Current Density

Volume current density, denoted as \(J_v\), is a measure of current per unit volume flowing through a material. Unlike surface current density, it details how current distributes through a three-dimensional volume and is expressed in amperes per cubic meter (A/m³).
In the conical conductor, the current flows radially outward from the vertex and spreads across the entire volume of the cone.
To calculate \(J_v\), you need the total current \(I\) and the volume \(V(r)\) that it traverses.

• The equation for the volume of a cone is \(V = \frac{1}{3}\pi R^2 h\), and thus \(J_v(r) = \frac{3I}{\pi h r^2 \tan^2(\theta)}\).

Volume current density provides insight into how currents distribute in three-dimensional space, which is vital for understanding conductivity and electromagnetic field interactions inside materials.

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that relates magnetic fields and electrical currents. It states that the line integral of the magnetic field \(\mathbf{H}\) around a closed path is equal to the total current enclosed by that path. Mathematically, it is expressed as: \[ \oint \mathbf{H} \cdot d\mathbf{l} = I_{enclosed} \]
This law is crucial for calculating magnetic fields in systems with symmetrical current distributions.
In the case of the cone and plane configuration, Ampere's Law allows us to determine the magnetic field \(\mathbf{H}(r, \theta)\) between and inside these structures.
The key steps are:

• Identifying the closed path (usually a loop where integration is practical).
• Calculating the current enclosed by the path.
• Solving for \(\mathbf{H}\) from the relationship \(H(r) = \frac{I_{enclosed}}{2\pi r}\).

Mastery of Ampere's Law is essential for anyone studying electromagnetic fields and currents, as it simplifies the computation of complex magnetic fields efficiently.