# Finding charge density

• ajith.mk91
In summary, there is a beam of electrons with a cross section of A in the space starting at x=0 and ending at x=20cm. The velocity of the electrons is a function of position x and is denoted as v(x). The goal is to calculate the volume charge density as a function of x. The suggested method is to take an imaginary plane perpendicular to the beam and use the equation of continuity to find the charge density, ρ. However, there is not enough information provided to solve the problem as the current density, J, is not specified. Additional information states that the current density is given at x=20cm as 10000A/m^2 and the goal is to find both the volume charge density

## Homework Statement

There is a beam of electrons in the space starting at x=0 and ending at x=20cm. The cross section of the beam is A.The velocity of the electrons is a function of position x and let it be v(x).Now i need to calculate the volume charge density as a function of x.

## The Attempt at a Solution

What i did is this:
I took an imaginary plane perpendicular to the beam.In a small time interval dt the amount of charge crossing the plane of area A is J.So dividing this J with v(x) should give the density. But J is not specified.Is there any other way to solve this?

I don't think there's enough information to solve the problem. Was that the complete problem as given to you?

The question has an additional info which i did not mention in my previous post.The current density is given at x=20cm as 10000A/m2(Actually a cathode is placed at x=0 and an anode is placed at x=20cm).But i was asked to find both the volume charge density and current density as a function of x.

The equation of continuity

$$\nabla\cdot\vec{J}+\frac{\partial\rho}{\partial t} = 0$$

There are a few different ways you could approach this problem. One way would be to use the formula for current density, J = nev, where n is the number density of electrons and e is the charge of an electron. You could then integrate this over the length of the beam to find the total current, and divide by the cross-sectional area A to get the current density. From there, you could use the definition of charge density, ρ = Q/V, where Q is the total charge and V is the volume, to calculate the charge density as a function of x.

Another approach could be to use the continuity equation, which states that the rate of change of charge density in a given volume is equal to the negative divergence of the current density. In this case, you could set up a differential equation for the charge density as a function of x and solve for it using the given information about the beam's velocity and cross-sectional area.

It's also worth noting that the problem statement does not specify any information about the number density of electrons or the current, so it may be difficult to calculate the charge density without making some assumptions or obtaining more information. As a scientist, it's important to carefully consider the available data and make reasonable assumptions or approximations when necessary.

## 1. What is charge density and why is it important in scientific research?

Charge density is a measure of the amount of electric charge per unit volume in a material. It is important in scientific research because it helps determine the behavior of charged particles and their interactions with other particles and fields.

## 2. How is charge density calculated?

Charge density is calculated by dividing the total amount of charge in a given volume by the volume itself. It is usually expressed in units of coulombs per cubic meter (C/m3) or coulombs per liter (C/L).

## 3. What factors affect charge density?

The factors that affect charge density include the type and amount of charge present, the size and shape of the charged particles, and the dielectric constant of the surrounding medium. Temperature and pressure can also have an impact on charge density.

## 4. How is charge density used in practical applications?

Charge density is used in a variety of practical applications, such as in the design of electrical circuits, the development of new materials for electronic devices, and the study of chemical reactions. It is also important in fields like electrochemistry, nanotechnology, and biophysics.

## 5. What are some techniques for measuring charge density?

There are several techniques for measuring charge density, including capacitance measurements, coulometric titration, and X-ray crystallography. Other methods, such as electrophoresis and electrorotation, can also be used to indirectly measure charge density in certain materials.