Question

The electron beam in a certain cathode ray tube possesses cylindrical symmetry, and the charge density is represented by \(\rho_{v}=-0.1 /\left(\rho^{2}+10^{-8}\right)\) \(\mathrm{pC} / \mathrm{m}^{3}\) for \(0<\rho<3 \times 10^{-4} \mathrm{~m}\), and \(\rho_{v}=0\) for \(\rho>3 \times 10^{-4} \mathrm{~m} .(a)\) Find the total charge per meter along the length of the beam; \((b)\) if the electron velocity is \(5 \times 10^{7} \mathrm{~m} / \mathrm{s}\), and with one ampere defined as \(1 \mathrm{C} / \mathrm{s}\), find the beam current.

Short answer

Answer: The total charge per meter along the length of the electron beam is approximately -3.35 x 10^-11 C/m, and the beam current is approximately -1.675 x 10^-3 A.

Step-by-step solution

Define the given variables

The given charge density is: $$ \rho_v = -\frac{0.1}{\rho^2 + 10^{-8}} \space \text{pC/m³} $$ \(\rho\) is the radial distance from the center of the beam which ranges from 0 to \(3 \times 10^{-4}\) m.

Set up the integration for finding the total charge per meter

The charge per unit length, \(Q'\), can be found by integrating the charge density over the cross-sectional area of the electron beam, and then multiply it by the length of the beam that is equal to 1 meter to get the total charge per meter. $$ Q' = \int_{0}^{2\pi}\int_{0}^{3 \times 10^{-4}} \rho_v \rho d\rho d\theta $$

Perform the integration

Substitute the value of \(\rho_v\) and then do proper integration: $$ Q' = -0.1 \int_{0}^{2\pi}d\theta \int_{0}^{3 \times 10^{-4}} \frac{\rho}{\rho^2 + 10^{-8}}d\rho $$ First, we integrate over \(\theta\): $$ Q' = -0.1 (2\pi) \int_{0}^{3 \times 10^{-4}} \frac{\rho}{\rho^2 + 10^{-8}}d\rho $$ Now, integrate over \(\rho\) using substitution method: let \(u = \rho^2 + 10^{-8}\), then \(du = 2\rho d\rho\). So, the integral becomes: $$ Q' = -0.1 (2\pi) \frac{1}{2}\int \frac{du}{u} $$ Now integrating it, we get: $$ Q' = -0.1\pi \ln(u) \big|_0^{3 \times 10^{-4}} = -0.1\pi \left( \ln\left( \left(3 \times 10^{-4}\right)^2 + 10^{-8}\right) - \ln\left(10^{-8}\right)\right) $$ Calculating the numeric value: $$ Q' \approx -3.35 \times 10^{-11} \space \text{C/m} $$

Calculate the beam current

Now we are given the electron velocity, \(v = 5 \times 10^7\space \text{m/s}\), and one ampere is defined as \(1\text{A} = 1\text{C/s}\). The beam current can be calculated by dividing the charge per meter by time it takes for one meter: $$ I = \frac{Q'}{t} = \frac{Q'}{1/v} = Q' \times v $$ Substituting the values, we get: $$ I = (-3.35 \times 10^{-11}\space \text{C/m}) \times (5 \times 10^7\space \text{m/s}) \approx -1.675 \times 10^{-3} \space \text{A} $$ So, the total charge per meter along the length of the beam is \(Q' \approx -3.35 \times 10^{-11} \space \text{C/m}\) and the beam current is \(I \approx -1.675 \times 10^{-3} \space \text{A}\).

Key concepts

Cathode Ray Tube Electromagnetics

A cathode ray tube (CRT) is a vacuum tube that uses an electron beam controlled by electric or magnetic fields to render visible images on a fluorescent screen. While old CRT televisions are rarely used today, the principles governing their operation are fundamental in electromagnetics and electronics.

The electron beam forms the core of CRT operation. Its behavior within the tube is governed by both electrostatic and magnetic forces. When high voltage is applied to the electron gun assembly, it accelerates electrons towards the screen. During their journey, the electrons can be steered by magnetic fields generated by deflection coils, allowing for precise control over the beam's impact point on the phosphorescent screen. Understanding this concept is essential when discussing the manipulation and utilization of cathode ray tubes in various applications.

Electrical Charge Density Calculation

Electrical charge density is a measure of how much electric charge is accumulated in a particular area and is essential for understanding electron beam characteristics in devices like CRTs. In our exercise, the charge density \( \rho_v \) is not evenly distributed; rather, it changes with the radius from the center of the beam, showing cylindrical symmetry.

To find the charge distribution, we typically integrate this density over the volume it occupies; for beams with cylindrical symmetry as in our scenario, we would use a cylindrical coordinate system for this integration. In applications, understanding how charge density varies with position can help in predicting the force on the charged particles and determining electric and magnetic fields inside electronic devices.

Beam Current Computation

Beam current is a crucial parameter in the operation of electron beams in devices like cathode ray tubes and particle accelerators. It quantifies the flow of electric charge along the beam per unit of time and is measured in amperes (A).

To compute the current within an electron beam, one usually multiplies the total charge per length unit \(Q'\) by the electron velocity \(v\). This is because an ampere of current is equal to a coulomb of charge passing a point per second, and in the case of a moving beam, this 'point' is the length of beam that the electrons traverse over time. High precision in the computation of the beam current is paramount, as it determines the brightness and focusing characteristics of the electron beam in a CRT.

Integration in Electromagnetic Problems

Integration serves as a powerful mathematical tool in solving electromagnetic problems, especially when dealing with distributed parameters, like charge density. In our example, calculating the total charge per meter along the electron beam required a double integration over the cross-sectional area.

Dealing with electromagnetic fields and forces often involves integrating over one, two, or three dimensions to find total charges, currents, or potentials. This is particularly true in electrostatic and magnetostatic situations where symmetries allow us to simplify the problems into more manageable forms. A clear understanding of integration techniques, including the substitution method used in our exercise, is vital for anyone studying electromagnetism or working on related technologies.