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Charge Density Formula

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What is Charge Density?

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The measure of electric charge accumulated in a particular field is called charge density. We can determine it in terms of volume, area, or length. We can divide the charge density formula into three types depending on its nature: (i) Linear (ii) charge density ( λ ) (iii) Surface charge density ( σ ) (iv) Volume charge density ( ρ ). Volume charge density is the quantity of charge per unit volume. Charge density measures the electric charge per unit measurement of the space. The space of measurement may be one, two, or three dimensional. Like mass density, charge density also varies with position. Thus, it can be positive or negative.


What is Linear Charge Density?

The quantity of charge per unit length, measured in coulombs per meter (cm−1), at any point on a line charge distribution, is called linear charge density (λ).

Suppose q is the charge and l is the length over which it flows, then the formula of linear charge density is λ= q/l, and the S.I. unit of linear charge density is coulombs per meter (cm−1).


Example: 

Q. A 50cm long thin rod has a total charge of 5mC uniformly distributed over it. What is the linear charge density?

Solution: q = 5 mC 

      = 5 × 10-3 C

l = 50 cm = 0.5 m 

             We have to find λ.

We know,

    λ = q / l  

       = 5 × 10-3 / 0.5

       = 10-2 c⋅m−1


What is Surface Charge Density?

The quantity of charge per unit area, measured in coulombs per square meter (Cm−2), at any point on a two-dimensional surface, is called the surface charge density(σ).

Suppose q is the charge and a is the area of the surface over which it flows, then the formula of surface charge density is σ = q/A, and the S.I. unit of surface charge density is coulombs per square meter (cm−2).


Example:

Q. A sphere has a charge of 12 C and radius 9 cm. Calculate the linear charge density?

Solution: Given,

Charge q = 12 C,

Radius r = 9 cm.

The surface charge density formula is given by,

σ = q / A

A=4 π r2

A = 4 π (0.09)2

A = 0.1017 m2

Surface charge density, σ = q / A

σ = 12 / 0.1017

  = 117.994

Therefore, σ = 117.994 cm−2


What is Volume Charge Density?

The quantity of charge per unit volume, at any point in a three-dimensional body, is called volume charge density(ρ).

Suppose q is the charge and V is the volume over which it flows, then the formula of volume charge density is ρ = q / V and the S.I. unit of volume charge density is coulombs per cubic meter (C⋅m−3)


Example

Q.A sphere of radius 1.85 cm has a charge of -260e spread through the volume uniformly. The sphere has a volume charge density of?

Solution: We are given,

The charge in the sphere, Q=−260e

The radius of the sphere, r=1.85cm

If Q is the total charge distributed over a volume V, then the volume charge density is given by the equation:

ρ= Q/V

The volume of a sphere:

V= 4/3πr3

The volume charge density of the sphere is:

ρ = Q / (4/3)πr3

  =−260e×3 / 4π(1.85cm)3

  =−9.8ecm−3

(Image to be added soon)


Solved Examples

1: Calculate the Charge Density of an Electric Field When a Charge of 6 C / m is Flowing through a Cube of Volume 3 m3.

Solution: Given the parameters are as follows,

Electric Charge, q = 6 C / m

Volume of the cube, V = 3 m3

The volume charge density formula is:

ρ = q / V

ρ =6 / 3

Charge density for volume ρ = 2C per m3.


2: Find the Volume Charge Density if the Charge of 10 C is Applied Across the Area of 2m3.

Solution: Given,

Charge q = 10 C

Volume v = 2 m3.

The volume charge density formula is,

ρ = q / v

ρ = 10C / 2m3

ρ = 5C/m3

FAQ (Frequently Asked Questions)

1. What are the Applications of Charge Density?

The various kinds of applications of charge density are given as follows:

  • It appears in the continuity equation for electric current and also appears in Maxwell's equations.

  • According to the electromagnetic field's principal source term, when the charge distribution moves, this corresponds to a current density.

  • The charge density of molecules affects both the chemical and separation parts of any process. It influences both metal-metal bonding and hydrogen bonding.

  • The charge density of ions influences their rejection by the membrane in processes such as nanofiltration.

  • Charge density, along with current density, is used in the concept of special relativity.

2. What is the Formula of Charge Density?

Charge density can be measured in terms of length, area, or volume depending on the body's dimension. It is of three types, as follows:

 

1. The quantity of charge per unit length, measured in coulombs per meter (Cm⁻¹), at any point on a line charge distribution, is called linear charge density ( λ ).

The formula of linear charge density is λ=q/l, such that q is the charge and l is the length of the body over which the charge is distributed.

 

2. The quantity of charge per unit area, measured in coulombs per square meter (Cm⁻²), at any point on a two-dimensional surface is called the surface charge density(σ).

The formula of surface charge density is σ=q/A, such that q is the charge and A is the area of the surface over which it flows.

 

3. The quantity of charge per unit volume, at any point in a three-dimensional body is called volume charge density ( ρ ).

The formula of volume charge density is ρ=q/v, such that q is the charge and v is the volume of the body over which it flows.