Question

A voltage source \(V_{0} \sin \omega t\) is connected between two concentric conducting spheres, \(r=a\) and \(r=b, b>a\), where the region between them is a material for which \(\epsilon=\epsilon_{r} \epsilon_{0}, \mu=\mu_{0}\), and \(\sigma=0 .\) Find the total displacement current through the dielectric and compare it with the source current as determined from the capacitance (Section \(6.3\) ) and circuit-analysis methods.

Short answer

In summary, the total displacement current through the dielectric material between two concentric spheres is equal in magnitude to the source current obtained using the capacitance method but flows in the opposite direction. This result can be derived by first finding the electric and magnetic fields in the dielectric region, calculating the displacement current density, and then comparing the resulting total displacement current with the source current obtained through the circuit-analysis method.

Step-by-step solution

Find the electric field in the dielectric region

To find the electric field inside the dielectric region between the conducting spheres, we will use the electric field due to a uniformly charged spherical capacitor. Since the voltage source is connected between the spheres, the electric field can be expressed as \(E = \frac{V_0 \sin \omega t}{r}\), where \(r\) is the distance from the center.

Use Maxwell's equation to find the magnetic field

Next, let us find the magnetic field in the dielectric region. Using the Ampere-Maxwell law in its integral form, \(\oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\), we find the magnetic field to be \(B = \mu_0 \epsilon_0 \omega V_0 \sin \omega t \frac{r}{2}\), where \(\omega\) is the angular frequency.

Calculate the displacement current density

Now we need to find the displacement current density, which can be calculated using \(J_D = \epsilon_0 \frac{d\vec{E}}{dt}\). Differentiate the electric field with respect to time to obtain \(J_D = -\epsilon_0 \omega V_0 \cos \omega t\).

Find the total displacement current

Now, integrate the displacement current density over the surface of the sphere with radius \(a\). Total displacement current can be found as \(I_D = \oint \vec{J_D} \cdot d\vec{A}\), which leads to \(I_D = -4\pi a^2 \epsilon_0 \omega V_0 \cos \omega t\).

Calculate the source current using capacitance

To find the source current using the capacitance method, first we determine the capacitance of the two concentric spheres: \(C = 4\pi\epsilon_0 \epsilon_r \frac{ab}{b-a}\). Then, find the source current by differentiating the voltage with respect to time and multiplying by the capacitance: \(I_C = C \times \frac{dV}{dt}\). We get \(I_C = 4\pi a^2 \epsilon_0 \epsilon_r \omega V_0 \cos \omega t\).

Compare the total displacement current with the source current

Finally, we need to compare the obtained total displacement current, \(I_D\), with the source current, \(I_C\) obtained using the capacitance method. It can be noticed that the expression for \(I_D\) is the same as the expression for \(I_C\) but with a negative sign. The negative sign means that the displacement current flows in the opposite direction of the source current, but the magnitudes are the same. So, the total displacement current through the dielectric is equal in magnitude to the source current obtained using the capacitance method.

Key concepts

Maxwell's Equations

Maxwell's equations are fundamental to understanding the behavior of electric and magnetic fields. They are a set of four equations that describe how electric charges and currents produce electric and magnetic fields, and how those fields interact with each other. The equations integrate electricity, magnetism, and light, forming the foundation of classical electrodynamics, optics, and electric circuits.

In essence, they are as follows:

• Gauss's law for electricity states that electric charges produce an electric field,
• Gauss's law for magnetism indicates that there are no magnetic monopoles,
• Faraday's law of induction describes how a changing magnetic field creates an electric field,
• and Ampere's law with Maxwell's addition explains how an electric current and a changing electric field produce a magnetic field.

Together, these principles explain a wide range of phenomena from radio broadcasts to the behavior of light itself.

Electric Field

The electric field is a vector field that surrounds electrically charged particles and time-varying magnetic fields. The strength and direction of the electric field at a particular point are given by the force that a small positive test charge would experience if placed at that point.

For a spherical capacitor, like in the exercise, the electric field between the spheres is determined by the charge distribution on the spheres and can be obtained from Gauss's law. The electric field in the exercise is directly proportional to the voltage applied and inversely proportional to the distance from the center, represented as \(E = \frac{V_0 \sin \omega t}{r}\). This field plays a crucial role in determining the displacement current through the dielectric.

Spherical Capacitor

A spherical capacitor is a type of capacitor that consists of two concentric spherical conductors, separated by a dielectric material. The capacitance of a spherical capacitor is influenced by the radii of the two spheres and by the permittivity of the dielectric material between them.

In the provided exercise, the region between the conducting spheres contains a dielectric material. The capacitance of such a configuration can be calculated using the formula \(C = 4\pi\epsilon_0 \epsilon_r \frac{ab}{b-a}\), where \(a\) and \(b\) represent the radii of the inner and outer spheres respectively, and \(\epsilon_r\) is the relative permittivity of the dielectric material. Capacitance measures a capacitor's ability to store an electric charge per unit voltage.

Ampere-Maxwell Law

The Ampere-Maxwell Law, an extension of Ampere's Law, adds an important component to the original law by incorporating the concept of displacement current. The term displacement current refers to the effect produced by a changing electric field in the dielectric material, even in the absence of a physical charge movement.

This law can be represented as \(\oint \vec{B} \cdot d\vec{l} = \mu_0 \left(I + \epsilon_0 \frac{d\Phi_E}{dt}\right)\), where \(\vec{B}\) is the magnetic field, \(\vec{l}\) is the length element of the closed loop, \(I\) is the conduction current, and \(\frac{d\Phi_E}{dt}\) is the rate of change of electric flux, representing the displacement current. In the exercise, the Ampere-Maxwell law enables us to relate the changing electric field between the capacitors with the magnetic field in the same region.

Dielectric Material

Dielectric materials are electrically insulating materials that can be polarized by an applied electric field. When a dielectric is placed between the plates of a capacitor, it increases the capacitor's capacitance by reducing the electric field strength for the same charge on the plates.

The effectiveness of a dielectric is characterized by its permittivity \(\epsilon\), which is the product of the vacuum permittivity \(\epsilon_0\) and the relative permittivity \(\epsilon_r\) of the material. High-permittivity dielectrics are preferred for capacitors as they allow more charge to be stored at a given voltage. In the context of the exercise, the dielectric material between the concentric spheres not only affects the capacitance but is also central to the concept of displacement current as it allows the electric field to change over time.