# Charge density of a disk with radius a in cylindrical coordinates

• hokhani
In summary, the conversation discusses the correct way to write the uniform charge density of a disk with radius a in cylindrical coordinates. One method involves using a constant A and a step function theta, while the other method involves using a delta function and a step function. The correct answer is given in spherical coordinates as well, but the routine method for solving such problems is not clear. The conversation also mentions a calculation for the total charge in the special case of a disk with finite height.
hokhani
To write the uniform charge density of a disk with radius a in cylindrical coordinates, If we do this form:
$\rho (x)=\frac{A\delta(z)\Theta (a-\rho)}{\rho}$ (A is constant that sholud be determined and $\theta$is step function), we get $A=\frac{Q}{2\pi a}$ and so:
$\rho (x)=\frac{\frac{Q}{2\pi a}\delta(z)\Theta (a-\rho)}{\rho}$
But we know that the correct one is:
$\rho (x)=Q\frac{\delta(z)\Theta (a-\rho)}{2\pi a^2}$.
Could anyone please tell me what is my mistake?

While I'm no expert on the subject, I DO notice that your "ro" term suddenly vanishes (term in the denominator).

Try starting with $\rho (x)=A\delta(z)\Theta (a-\rho)$

dauto said:
Try starting with $\rho (x)=A\delta(z)\Theta (a-\rho)$

Ok, in this special case it gives the correct answer, but in the spherical coordinate system it doesn't give the correct answer. My principle question is what is the routine way of solving such problems?

Take a cylindrical disk of finite height filled uniformly with a total charge $Q$. Obviously the charge density is given by
$$\rho=\frac{Q}{\pi a^2 h} \Theta(a-r_{\perp}) \Theta(-h/2<z<h/2).$$
I write $r_{\perp}$ for the radial coordinate in order to avoid conflicts with $\rho$ as the symbol for the charge density. In the limit $h \rightarrow 0^+$ this gives
$$\rho=\frac{Q}{\pi a^2} \Theta(a-r_{\perp}) \delta(z).$$
You can easily check that this gives you the correct total charge,
$$\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \rho(\vec{x})=\int_{0}^{a} \mathrm{d} r_{\perp} \int_0^{2 \pi} \mathrm{d} \varphi \int_{\mathbb{R}} \mathrm{d}z r_{\perp} \frac{Q}{\pi a^2} \delta(z) = \frac{Q}{\pi a^2} 2 \pi \frac{a^2}{2}=Q.$$

hokhani said:
Ok, in this special case it gives the correct answer, but in the spherical coordinate system it doesn't give the correct answer. My principle question is what is the routine way of solving such problems?

What do you mean it doesn't work in spherical coordinates? If you post your calculation I might be able to comment.

## 1. What is the definition of charge density in cylindrical coordinates?

Charge density in cylindrical coordinates refers to the amount of electric charge per unit area on a disk with radius a. It is denoted by the symbol ρ and has units of coulombs per square meter (C/m²).

## 2. How is charge density related to the electric field on a disk with radius a?

The electric field on a disk with radius a is directly proportional to the charge density. This means that as the charge density increases, the electric field also increases. The relationship is represented by the equation E = ρ/ε₀, where E is the electric field, ρ is the charge density, and ε₀ is the permittivity of free space.

## 3. What is the formula for calculating charge density on a disk with radius a?

The formula for calculating charge density on a disk with radius a is ρ = Q/A, where Q is the total charge on the disk and A is the area of the disk. This formula assumes that the charge is evenly distributed on the disk.

## 4. How does the charge density affect the electric potential on a disk with radius a?

The charge density on a disk with radius a affects the electric potential by creating a potential difference between points on the disk. The higher the charge density, the greater the potential difference. This potential difference is calculated using the formula V = kQ/r, where V is the potential difference, k is the Coulomb's constant, Q is the total charge on the disk, and r is the distance from the center of the disk.

## 5. How can the charge density on a disk with radius a be measured experimentally?

The charge density on a disk with radius a can be measured experimentally by using a device called an electric field sensor. This sensor measures the strength of the electric field at different points on the disk and can be used to calculate the charge density using the formula E = ρ/ε₀. Other methods such as Coulomb's law and Gauss's law can also be used to indirectly calculate the charge density on the disk.

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