Question

A charged-particle beam is used to inject a charge, \(Q_{0}\), 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

In this problem, we derived a differential equation for the charge in the injection region as a function of time, using the continuity equation and Ohm's law. After solving the differential equation, we found the charge function \(Q(t)\). We then used conductivity and permittivity values for copper and quartz to calculate the time it takes for the charge to decrease by half. The results show that it takes approximately \(1.03 \times 10^{-19}\) seconds for the charge to decrease by half in copper, and \(24.6\) seconds for the charge to decrease by half in quartz.

Step-by-step solution

Differential Equation for Charge

To derive a differential equation for the charge \(Q(t)\) in the injection region, we need to consider the current density \(J\) due to the charge motion. As the charge moves from the irregular region to the outer surface of the block, it creates a current. This current density is given by Ohm's law: \(J = \sigma E,\) where \(\sigma\) is the conductivity of the material and \(E\) is the electric field created by the charge on the region. We can also write the electric field in terms of electric displacement field \(D\), which is given by: \(D = \epsilon E.\) Combining both equations, we can rewrite current density as: \(J = \frac{\sigma}{\epsilon}D.\) Now, we can consider the continuity equation that relates divergence of current density \(\nabla \cdot J\) to rate of charge decay in the region. The continuity equation is given by: \(\nabla \cdot J = -\frac{dQ}{dt},\) where \(\frac{dQ}{dt}\) represents the rate of charge decay. Substituting \(J\) into the continuity equation and considering that \(\nabla \cdot D = Q\), we get: \(\nabla \cdot \left(\frac{\sigma}{\epsilon}D\right) = -\frac{dQ}{dt}.\) Since \(\nabla \cdot D = Q\), we can rewrite this equation as: \(\frac{\sigma}{\epsilon}\nabla \cdot D = -\frac{dQ}{dt}.\) This is the differential equation for the charge \(Q(t)\) in the injection region as a function of time.

Solving the Differential Equation for Charge

Rearrange the differential equation to separate variables: \(\frac{dQ}{Q} = -\frac{\sigma }{\epsilon} dt.\) Now integrate both sides: \(\int_{Q_0}^Q \frac{dQ'}{Q'} = -\frac{\sigma}{\epsilon} \int_0^t dt'.\) Applying the integral rules: \(\ln \left(\frac{Q}{Q_0}\right) = -\frac{\sigma}{\epsilon} t.\) Now, solve for \(Q(t)\): \(Q(t) = Q_0 e^{-\frac{\sigma}{\epsilon}t}.\) This is the function that describes the charge \(Q(t)\) in the injection region for all \(t \geq 0\).

Calculating the Time for Charge to Decrease by Half for Copper and Quartz

Now, we need to calculate the time for the charge in the injection region to decrease by half, i.e., when \(Q(t) = \frac{Q_0}{2}\). To do this, we will use the conductivity and permittivity of copper and quartz. For copper: \(\sigma_{Cu} \approx 5.96 \times 10^7 S/m\) \(\epsilon_{Cu} \approx \epsilon_0 = 8.85 \times 10^{-12} F/m\) For quartz (crystalline \(\text{SiO}_2\)): \(\sigma_{q} \approx 1 \times 10^{-18} S/m\) \(\epsilon_{q} \approx 4 \epsilon_0 = 4 \times 8.85 \times 10^{-12} F/m\) Now, let's find the time \(t_{1/2}\) for which \(Q(t_{1/2}) = \frac{Q_0}{2}\): \(\frac{Q_0}{2} = Q_0 e^{-\frac{\sigma}{\epsilon}t_{1/2}}.\) Solve for \(t_{1/2}\): \(t_{1/2} = \frac{\epsilon}{\sigma} \ln 2.\) For copper, \(t_{1/2, Cu} = \frac{8.85 \times 10^{-12}}{5.96 \times 10^7} \ln 2 \approx 1.03 \times 10^{-19}~s.\) For quartz, \(t_{1/2, q} = \frac{4 \times 8.85 \times 10^{-12}}{1 \times 10^{-18}} \ln 2 \approx 24.6 ~s.\) The time for the charge in the injection region to decrease by half is approximately \(1.03 \times 10^{-19}\) seconds for copper and \(24.6\) seconds for quartz.

Key concepts

Current Density

When we talk about current density, we're referring to the amount of electric current flowing per unit area through a material. It's a vector quantity, meaning it has both magnitude and direction, and is usually denoted by the symbol J. In a charged-particle beam within an ohmic material like the one in our exercise, current density arises from the movement of charges from a region of high concentration to one of lower concentration, aiming to balance the distribution of charges.

Current density is crucial because it helps us understand how intensely current flows in different parts of a conductor. It impacts various electrical phenomena, including the strength of magnetic fields around conductors, the heat generated by the flow of current, and the efficiency of energy transfer in electrical devices.

• Mathematically, it's expressed as J = I/A, where I is the electric current and A is the cross-sectional area through which the current flows.
• In the context of the problem, knowing the current density allows us to derive equations related to the charge decay over time.

Ohm's Law

Ohm's Law is a foundational principle in electrical engineering and physics. This law tells us that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and is inversely proportional to the resistance of the conductor. The mathematical expression for Ohm's law is V = IR, where V is the voltage (potential difference), I is the current, and R is the resistance.

Ohm's law is essential for understanding how to control the flow of electric current in circuits. It helps designers create circuits that can handle specific voltages and currents without overheating or failing. For conductive materials, such as the ohmic material in our exercise:

• The relationship between current density (J) and electric field (E) is given by J = σE, where σ represents the material's conductivity.
• This simplifies to E = J / σ, allowing engineers to predict the current's behavior in response to changes in material properties or electric field strength.

Electric Displacement Field

The electric displacement field, denoted as D, is related to the electric field E and provides a picture of how an electric field affects the distribution of electric charges in materials, especially within dielectrics (insulators). It's essential for analyzing the behavior of materials in an electric field and for designing capacitors and other components used in electronics. The field D considers both the free charges (which respond to external fields) and the bound charges (which are associated with the atomic structure of the material).

The electric displacement field is directly proportional to the electric field and is given by the product of the permittivity of the material ε (a measure of how much electric field is 'allowed' through the material) and the electric field itself, D = εE.

• For the charged-particle beam problem, the electric displacement field helps relate the electric field in the material to the current density and charge decay over time.
• This concept is encompassed in the equation J = σE = (σ/ε)D, combing both Ohm's law and the definition of electric displacement.

Continuity Equation

The continuity equation is a fundamental principle in physics and engineering that deals with the conservation of some quantity. In the context of electric charge, it states that the rate at which charge density decreases within a given volume is equal to the net current's divergence (the net flux of current) out of the volume. This concept is indispensable when analyzing systems where the charge distribution changes over time, such as in our exercise where charge moves from an injection region to the surface.

The continuity equation for electrical charge is expressed as ∇·J = -dQ/dt, where ∇·J is the divergence of current density and dQ/dt is the rate of change of charge. This equation ensures that:

• Charge is conserved within the system.
• We can link the flow of electric currents to the temporal variation of charge density.

Differential Equation for Charge Decay

The differential equation for charge decay describes how the charge within a particular region changes over time. For the charged-particle beam example, we derived a differential equation (σ/ε∇·D = -dQ/dt) to model the behavior of charge as it moves from the region where it was injected toward the outer surface of the block. By solving this equation, we can predict how quickly the charge will decay and ultimately vanish or become uniformly distributed.

This equation is a cornerstone for analyzing charge redistribution in materials, designing capacitors, batteries, and even understanding natural phenomena such as lightning. Its solutions allow us to determine lifetimes of charge carriers and rates of charge dissipation, which are critical for both theoretical and applied sciences.

Conductivity and Permittivity Relations

In materials science and electrodynamics, the concepts of conductivity and permittivity are interconnected and play pivotal roles in understanding the behavior of materials under electric fields. Conductivity, denoted as σ, measures a material's ability to conduct electric current. In contrast, permittivity, denoted as ε, measures a material's ability to resist an electric field (its capacity to be polarized by the field).

In our charged-particle beam problem, the relation between the two can be seen in the derived formula: σ/ε appears as a ratio in the exponential part of the solution Q(t) = Q_0 e^{(-σ/ε)t}. A high conductivity to permittivity ratio indicates rapid charge decay, while a low ratio suggests a slower rate

• A conducting material like copper has high conductivity but relatively low permittivity.
• An insulating material such as quartz has low conductivity and higher relative permittivity. This contrast leads to a marked difference in how quickly the charge decreases by half in the two materials, as shown in the steps for calculating t_{1/2}.

Time for Charge to Decrease by Half

The time for charge to decrease by half, often symbolized as t_{1/2}, is a specific instance of a broader concept called the relaxation time. It's the time required for a quantity that decreases exponentially to reach half of its initial value. In electrical circuits, this concept is vital in understanding transient behaviors - how a system responds to changes from one state to another.

From the solution for the charge decay Q(t) = Q_0 e^{(-σ/ε)t}, setting Q(t) = Q_{0}/2 and solving for time gives us t_{1/2} = (ε/σ) ln 2. It's noteworthy that the time to halve the charge directly depends on the materials' properties:

• Materials with high electrical conductivity and low permittivity, like copper, have very short t_{1/2}, meaning they neutralize injected charges almost instantaneously.
• In contrast, materials with low conductivity and higher permittivity, such as quartz, have much longer t_{1/2}, keeping injected charges for extended periods.

This concept is especially important in designing and analyzing electronic components such as capacitors and memory devices where charge retention is critical.