Question

If volume charge density is given as ${\rho }_v=(\cos \omega t)/r^2\mathrm{C}/{\mathrm{m}}^2$ in spherical coordinates, find $\mathbf{J}$. It is reasonable to assume that $\mathbf{J}$ is not a function of $\theta$ or $\varphi$.

Short answer

The current density vector $\mathbf{J}$ can be found given the volume charge density ${\rho }_v=(\cos \omega t)/r^2\mathrm{C}/{\mathrm{m}}^2$ in spherical coordinates, after integrating and solving for the continuity equation. The final expression for the current density vector is given by: $\mathbf{J}=-\frac{\omega \sin \omega t}{r^3}\hat{\mathbf{r}}$, indicating that the current density depends on the radial distance and time.

Step-by-step solution

Understand the main components of the problem

Given the volume charge density ${\rho }_v=(\cos \omega t)/r^2\mathrm{C}/{\mathrm{m}}^2$ in spherical coordinates, we are tasked with finding the current density vector $\mathbf{J}$. Since ${\rho }_v$ is given in spherical coordinates, the current density vector $\mathbf{J}$ will also be in spherical coordinates, which can be expressed as $\mathbf{J}=J_r\hat{\mathbf{r}}+J_{\theta }\hat{\theta }+J_{\varphi }\hat{\varphi }$.

Determine radial current density $J_r$

Since $\mathbf{J}$ does not have angular dependence (not a function of $\theta$ or $\varphi$), we only need to find the radial component $J_r$. To do so, we use the continuity equation, which is $\mathrm{\nabla }\cdot \mathbf{J}=-\frac{\partial {\rho }_v}{\partial t}$. The continuity equation in spherical coordinates for the radial component is: $\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{J}_r)=-\frac{\partial {\rho }_v}{\partial t}$ Now, evaluate $\frac{\partial {\rho }_v}{\partial t}$: $\frac{\partial {\rho }_v}{\partial t}=-\frac{\omega \sin \omega t}{r^2}\mathrm{C}/{\mathrm{m}}^2$ Substitute this into the continuity equation: $\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{J}_r)=\frac{\omega \sin \omega t}{r^2}\mathrm{C}/{\mathrm{m}}^2$

Solve for $J_r$

To find $J_r$, we need to integrate both sides with respect to $r$. Focus on the left side, then the right side: (Left Side) $\int \frac{1}{r^2}\frac{\partial }{\partial r}(r^2{J}_r)dr=\int \frac{\partial }{\partial r}(r^2{J}_r)dr$ $⟹$ $\Rightarrow$ $r^2{J}_r=F_1(t)+C_1$ (Right Side) $\int \frac{\omega \sin \omega t}{r^2}dr=\omega \sin \omega t\int \frac{1}{r^2}dr$ $⟹$ $\Rightarrow$ $-\frac{\omega \sin \omega t}{r}=F_2(t)+C_2$ Now equating the left side with the right side: $r^2{J}_r=-\frac{\omega \sin \omega t}{r}+C$ Solve for $J_r$: $J_r=-\frac{\omega \sin \omega t}{r^3}+\frac{C}{r^2}$

Write the final current density vector $\mathbf{J}$

Now that we have determined the radial component $J_r$, we can write the final expression for the current density vector in spherical coordinates. The other components, $J_{\theta }$ and $J_{\varphi }$ are zero, so: $\mathbf{J}=-\frac{\omega \sin \omega t}{r^3}\hat{\mathbf{r}}$

Key concepts

Volume Charge Density

Volume charge density, often symbolized as ${\rho }_v$, represents the amount of electric charge per unit volume in a given region of space. In the context of electromagnetics, it is crucial in determining how charges are distributed in a medium. The volume charge density can vary with time as well as spatial location. In this particular exercise, ${\rho }_v$ is given by the expression ${\rho }_v=(\cos \omega t)/r^2\mathrm{C}/{\mathrm{m}}^2$. This relationship implies that the charge density changes over time due to the cosine term, and it also depends on the radial distance $r$ from a point in space, decaying with the square of the distance.

Understanding how volume charge density impacts electric fields is crucial. It serves as a fundamental component in the laws governing electromagnetics, such as Gauss's Law, which relates electric fields to the distribution of charges. Volume charge density is commonly used in solving problems related to electric potential and fields in dielectric materials.

Current Density Vector

The current density vector, denoted as $\mathbf{J}$, describes the flow of electric charge through a given area. It is commonly measured in amperes per square meter (A/m²). The current density vector is pivotal when understanding how electric currents travel through materials. Its components depend on the coordinate system being used.

For this problem, we focus on spherical coordinates, which implies $\mathbf{J}$ can be expressed as $\mathbf{J}=J_r\hat{\mathbf{r}}+J_{\theta }\hat{\theta }+J_{\varphi }\hat{\varphi }$. Since the problem states that J has no angular dependence, the angular components $J_{\theta }$ and $J_{\varphi }$ are zero. Thus, we're left to determine only the radial current density component $J_r$.

Applying concepts like the continuity equation helps establish a relationship between changes in charge density and current density, solidifying our understanding of how charge conservation laws apply to dynamic systems.

Spherical Coordinates

Spherical coordinates are a system of curvilinear coordinates that extend polar coordinates into three dimensions, making them ideal for problems with spherical symmetry. This system is defined by the radial distance $r$, the polar angle $\theta$ (commonly measured from the positive z-axis), and the azimuthal angle $\varphi$ (measured in the xy-plane from the positive x-axis).

In the given exercise, the volume charge density is represented in spherical coordinates, which is practical for problems involving radial symmetry like fields around a sphere or point charge. The current density vector, being in the same coordinate system, is naturally expressed as $\mathbf{J}=J_r\hat{r}+J_{\theta }\hat{\theta }+J_{\varphi }\hat{\varphi }$. This coordinate system simplifies the mathematics involved in calculating fields and densities by aligning with the natural symmetry of the problem.

Using spherical coordinates makes calculating differential areas and volumes more intuitive in radially symmetric systems, as factors of $r\sin \theta$ appear in the equations, accounting for changes over the surface of a sphere or shell.

Continuity Equation

The continuity equation is a fundamental principle in electromagnetics that represents the conservation of electric charge. It mathematically ensures that electric charges neither spontaneously appear nor disappear. In its differential form, it is expressed as $abla\cdot \mathbf{J}=-\frac{\partial {\rho }_v}{\partial t}$. This equation states that the divergence of the current density $\mathbf{J}$ is equal to the negative rate of change of the volume charge density ${\rho }_v$ with respect to time.

For the radial component $J_r$ in spherical coordinates, the equation modifies slightly to account for the geometry, becoming $\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{J}_r)=-\frac{\partial {\rho }_v}{\partial t}$. This ensures the radial flow of current density accounts for conservation within a spherical volume.

The continuity equation is essential for solving problems where charge and current densities evolve over time. It forms the basis for understanding dynamic electromagnetic systems and lays the groundwork for more advanced concepts in fields like circuit design and electromagnetic wave propagation.