Volume charge density – formula for a Sphere, Cylinder

A three dimensional conductor can have three types of charge distributions. These are Line charge distribution, surface charge distribution and volume charge distribution. Obviously, there will be three types of charge densities for these three distributions. In another article, we have discussed the Linear charge density and surface charge density of different conductors. In this article, we are going to discuss the volume distribution of charge and volume charge density formula of different conductors like a sphere, a cylinder, etc.

• Volume charge distribution
• Volume density of charge
• Symbol of Volume charge density
• Equation of Volume charge density
• Volume charge density unit
• Dimension of volume charge density
• Volume charge density formula of different conductors
• Integral equation of charge density and charge

Volume charge distribution

In a three dimensional (3D) conductor, electric charges can be present inside its volume. This type of distribution of electric charge inside the volume of a conductor is known as the volume charge distribution. A spherical conductor, a cylindrical conductor, etc. can have volume charge distribution.

What is volume charge density?

The volume charge density of a conductor is defined as the amount of charge stored per unit volume of the conductor. Only the conductors with three dimensional (3D) shapes like a sphere, cylinder, cone, etc. can have volume charge density.

Symbol of Volume charge density

The volume density of charge is represented by the Greek letter rho ($\color{Blue}\rho$).

Volume charge density equation

If Q be the amount of charge inside a volume V of a conductor, then the formula for volume charge density of the conductor is, $\color{Blue}\rho=\frac{Q}{V}$…….(1)

This is the fundamental equation of volume density of electric charge. Since, the formula of volume is different for different shapes, the formula of volume charge density also has different forms for conductors of different shapes. These formulae are at below.

Volume charge density unit

SI unit of electric charge is Coulomb (C) and of volume is m3. Therefore, the SI unit of volume density of charge is C.m-3 and the CGS unit is StatC.cm-3.

Dimension of Volume charge density

The dimension of electric charge is [TI] and the dimension of volume is [L3]. Then, the dimensional formula of volume charge density is [L-3TI].

Volume charge density formula of different conductors

As the volume formula is different for the conductors of different shapes, therefore we can get different forms for the volume charge density formula for different shapes.

Volume charge density of sphere

If a spherical conductor of radius R contains Q amount of charge inside its volume, then the formula for the volume charge density of the sphere is, $\color{Blue}\rho=\frac{Q}{\frac{4}{3}\pi R^{3}}$………(2)

Volume charge density of cylinder

If a cylindrical conductor of length L and radius r contains Q amount of charge inside its volume, then the formula of volume charge density of the cylinder is $\color{Blue}\rho=\frac{Q}{\pi r^{2} L}$……..(3)

Integral relation between charge and volume charge density

From equation-(1), electric charge = volume charge density × volume

The integral form of this relation is, $\color{Blue}Q = \int\rho .dV$……..(4)

This is all from this article on charge density of a conductor inside its volume. If you have any doubt on this topic you can ask me in the comment section.

Thank you!

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