Question

A charged-particle beam is used to inject a charge, \(Q\), into a small, irregularly shaped region (not a cavity, just some region within the solid block) in the interior of a block of ohmic material with conductivity \(\sigma\) and permittivity \(\epsilon\) at time \(t=0 .\) Eventually, all the injected charge will move to the outer surface of the block, but how quickly? a) Derive a differential equation for the charge, \(Q(t),\) in the injection region as a function of time. b) Solve the equation from part (a) to find \(Q(t)\) for all \(t \geq 0\). c) For copper, a good conductor, and for quartz (crystalline \(\mathrm{SiO}_{2}\) ), an insulator, calculate the time for the charge in the injection region to decrease by half. Look up the necessary values. Assume that the effective "dielectric constant" of copper is \(1.00000 .\)

Short answer

Answer: The half-times for charge to decrease by half in the injection region are approximately \(10.5\times 10^{-20}\ \text{s}\) for copper and \(21.8\times 10^6\ \text{s}\) for quartz.

Step-by-step solution

(a) Derive the differential equation

Let's start by writing down Gauss's law for this problem: \[ \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon} \] Here, \(\mathbf{E}\) is the electric field, \(d\mathbf{A}\) is the differential area element, and \(Q\) is the time-varying charge inside the region. Now, recall Ohm's law \(\mathbf{J} = \sigma \mathbf{E}\), where \(\mathbf{J}\) is the current density, and \(\sigma\) is the conductivity of the material. Thus the current leaving the region is given by \[ I(t) = \oint \mathbf{J} \cdot d\mathbf{A} = \oint \sigma \mathbf{E} \cdot d\mathbf{A} \] Now substituting Gauss's law into this expression we get, \[ I(t) = \frac{\sigma Q(t)}{\epsilon} \] Since \(I(t) = -\frac{dQ(t)}{dt}\), we can write the differential equation as: \[ -\frac{dQ(t)}{dt} = \frac{\sigma Q(t)}{\epsilon} \]

(b) Solve the differential equation

To solve the given first-order linear differential equation, we can first separate the variables: \[ \frac{dQ(t)}{Q(t)} = -\frac{\sigma}{\epsilon} dt \] Integrating both sides, we get: \[ \int \frac{dQ(t)}{Q(t)} = -\int \frac{\sigma}{\epsilon} dt \] Which leads to: \[ \ln(Q(t)) = -\frac{\sigma}{\epsilon}t + C \] Taking the exponent of both sides, we have \[ Q(t) = Q_0 e^{-\frac{\sigma}{\epsilon}t} \] where \(Q_0\) is the initial charge in the region at time \(t = 0\).

(c) Calculate the half-time for copper and quartz

Now, we want to find the time \(t_{1/2}\) for the charge in the injection region to decrease by half for copper and quartz, i.e., when \(Q(t_{1/2}) = \frac{1}{2}Q_0\). Using the expression for \(Q(t)\), we can write \[ \frac{1}{2}Q_0 = Q_0 e^{-\frac{\sigma}{\epsilon}t_{1/2}} \] Dividing both sides by \(Q_0\), and taking the natural logarithm of both sides, we have \[ \ln(\frac{1}{2}) = -\frac{\sigma}{\epsilon}t_{1/2} \] So, \[ t_{1/2} = \frac{\epsilon}{\sigma}\ln(2) \] We need to find \(t_{1/2}\) for copper and quartz. Look up the given values for \(\sigma\) and \(\epsilon\) for copper and quartz, and assume that the effective dielectric constant of copper is \(1.00000\). For copper, \(\sigma = 5.8\times 10^7\ \text{S}\cdot \text{m}^{-1}\) and \(\epsilon = 8.854 \times 10^{-12}\ \text{F}\cdot \text{m}^{-1}\). We will use these values to get the half-time for copper: \[ t_{1/2,\text{copper}} = \frac{8.854\times10^{-12}\ \text{F}\cdot \text{m}^{-1}}{5.8\times 10^7\ \text{S}\cdot \text{m}^{-1}}\ln(2) \approx 10.5\times 10^{-20}\ \text{s} \] For quartz, \(\sigma = 10^{-18}\ \text{S}\cdot \text{m}^{-1}\) and \(\epsilon = 3.78 \times 8.854\times 10^{-12}\ \text{F}\cdot\text{m}^{-1}\) (\(3.78\) is the dielectric constant of quartz). We get the half-time for quartz as: \[ t_{1/2,\text{quartz}} = \frac{3.78 \times 8.854\times 10^{-12}\ \text{F}\cdot \text{m}^{-1}}{10^{-18}\ \text{S}\cdot\text{m}^{-1}}\ln(2) \approx 21.8\times 10^6\ \text{s} \] So, the time for the charge in the injection region to decrease by half is approximately \(10.5\times 10^{-20}\ \text{s}\) for copper and \(21.8\times 10^6\ \text{s}\) for quartz.

Key concepts

Gauss's Law

Gauss's law forms the foundation of electrostatics, relating the electric flux flowing out of a closed surface to the charge enclosed by that surface. Mathematically, it's expressed as \[ \oint \mathbf{E} \bullet d\mathbf{A} = \frac{Q}{\epsilon} \] with \(\mathbf{E}\) representing the electric field, \(d\mathbf{A}\) the infinitesimal area of the surface, and \(Q\) the charge within.
To practically apply Gauss's law in an exercise like the one we're discussing, we need to understand that the law allows us to calculate the electric field produced by a given charge distribution. As part of understanding charge decay in conductors and insulators, Gauss's law is utilized in conjunction with Ohm's law to derive the rate at which charge dissipates over time.

Ohm's Law

Ohm’s law is a fundamental principle in the field of electronics and electromagnetism, which states that the current passing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. Introducing it as \(I = \frac{V}{R}\) for a basic circuit, or, when it comes to continuous media like the material of our problem, as \(\mathbf{J} = \sigma \mathbf{E}\), with \(\mathbf{J}\) being the current density, \(\sigma\) the electrical conductivity, and \(\mathbf{E}\) the electric field. This equation is key to understanding how an electric current flows through materials and is crucial for analyzing the charge decay in the exercise.

Differential Equation

Differential equations are mathematical equations that involve functions and their derivatives. They are used to describe various physical phenomena, such as the rate of change of quantities. In this context, we derive a differential equation to model the decay of charge in a material, which looks like \[ -\frac{dQ(t)}{dt} = \frac{\sigma Q(t)}{\epsilon} \]. Solving this equation helps us understand how the charge within a material changes over time. These equations are potent tools because they not only describe the change but also predict future behavior under given conditions.

Charge Decay Time

Charge decay time is a measure of how quickly charge reduces over time in a material. It is particularly relevant when discussing semiconductors, conductors, and insulators. The exercise provided deals with finding the time required for the charge to reduce by half, often termed as 'half-life'. By solving the derived differential equation, we can quantify this decay and thereby understand the effectiveness and properties of the material in question. Knowing the charge decay time is foundational in the design and application of various electronic devices.

Electrical Conductivity

Electrical conductivity, denoted by \(\sigma\), is a measure of a material's ability to conduct an electric current. It appears in Ohm’s law for continuous media and is central to the concept of charge decay in conductors and insulators. It enables us to understand why materials like copper have very short charge decay times, as they have high conductivity, while insulators like quartz have much longer decay times due to their extremely low conductivity. Electrical conductivity is a property that radically affects the behavior of charge within different materials.

Permittivity

Permittivity, denoted by \(\epsilon\), is an essential property of a material that affects how it transmits electric fields. In the context of Gauss's law, \(\epsilon\) lowers the electric field for a given charge. In materials with high permittivity, charges tend to remain longer, indicating a slow decay, mainly in insulators. For conductors, which have a low permittivity, charges move quickly to the surface. The permittivity of free space, \(\epsilon_0\), plays a role in defining the unit of charge in the International System of Units (SI), influencing our calculations and understanding of electric phenomena.

Ohmic Material

An ohmic material is one that conforms to Ohm’s law, meaning its resistance remains constant as the electric current through it varies, provided the physical conditions do not change. In the exercise, the block of material into which charge is injected is assumed to be ohmic. This assumption simplifies the analysis of charge decay as it implies a linear relationship between the current and the electric field within the material, and allows the use of the simple form \(\mathbf{J} = \sigma \mathbf{E}\) to proceed with solving the problem.

Dielectric Constant

The dielectric constant, also known as the relative permittivity, is the ratio of the permittivity of a substance to the permittivity of free space \(\epsilon_0\). It is a dimensionless number that measures a material's ability to resist an electric field, contributing to its capacitance when used as a dielectric in a capacitor. The dielectric constant impacts the charge distribution within materials – higher values signify greater ability to store charge. This is why in our exercise, we need to adjust for the dielectric constant of quartz to accurately calculate the charge decay time.