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Charge Density Formula

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  • Last Updated : 24 May, 2022
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When measuring electric fields from various continuous charge distributions such as linear, surface, and volume, we come across electric charge density. When understanding current electricity, we must also consider the concept of charge density. To understand charge density, we must first understand this concept of density. The density of an object is defined as its mass per unit volume. Similarly, depending on the type of continuous charge arrangement, we can think of charge density as charge per unit length, surface, or volume.

What is Charge density?

Charge density is defined as the amount of electric charge that can be accumulated over a unit length or unit area or unit volume of a conductor. In other words, it indicates how much charge is stored in a specific field. It calculates the distribution of the charge and can be positive or negative. 

The charge may be scattered over a one-dimensional or two-dimensional or three-dimensional surface. The charge density is categorized into three types: 

  1. Linear charge density
  2. Surface charge density, and 
  3. Volume charge density. 

Its value is directly proportional to the amount of charge but changes inversely with the surface dimensions.

Linear charge density

The linear charge density is defined as the amount of charge present over a unit length of the conductor. It is denoted by the symbol lambda (λ). Its standard unit of measurement is Coulombs per meter (Cm-1) and the dimensional formula is given by [M0L-1T1I1]. 

Its formula equals the ratio of charge value to the length of the conducting surface.

λ = q/l

where,

  • λ is the linear charge density,
  • q is the charge,
  • l is the length of surface.

Surface charge density

The surface charge density is defined as the amount of charge present over a unit area of the conductor. It is denoted by the symbol sigma (σ). Its standard unit of measurement is coulombs per square meter (Cm-2) and the dimensional formula is given by [M0L-2T1I1]. 

Its formula equals the ratio of charge value to the area of the conducting surface.

σ = q/A

where,

  • σ is the surface charge density,
  • q is the charge,
  • A is the area of surface.

Volume charge density

The volume charge density is defined as the amount of charge present over a unit volume of the conductor. It is denoted by the symbol rho (ρ). Its standard unit of measurement is coulombs per cubic meter (Cm-3) and the dimensional formula is given by [M0L-3T1I1]. 

Its formula equals the ratio of charge value to the volume of the conducting surface.

ρ = q/V

where,

  • σ is the surface charge density,
  • q is the charge,
  • V is the volume of surface.

Sample Problems

Problem 1: Calculate the linear charge density of a surface if the charge is 2 C and length is 4 m. 

Solution:

We have,

q = 2 

l = 4

Using the formula we have,

λ = q/l

= 2/4

= 0.5 Cm-1

Problem 2: Calculate the linear charge density of a surface if the charge is 5 C and the length is 3 m.

Solution:

We have,

q = 5

l = 3

Using the formula we have,

λ = q/l

= 5/3

= 1.67 Cm-1

Problem 3: Calculate the charge if the linear charge density of a surface is 3 Cm-1 and the length is 5 m.

Solution:

We have,

λ = 3

l = 5

Using the formula we have,

λ = q/l

=> q = λl

= 3 (5)

= 15 C

Problem 4: Calculate the surface charge density of a surface if the charge is 20 C and the area is 10 m2.

Solution:

We have,

q = 20

A = 10

Using the formula we have,

σ = q/A

= 20/10

= 2 Cm-2

Problem 5: Calculate the charge if surface charge density of a surface is 5 Cm-2 and the area is 20 m2.

Solution:

We have,

σ = 5 

A = 20

Using the formula we have,

σ = q/A

=> q = σA

= 5 (20)

= 100 C

Problem 6: Calculate the volume charge density of a surface if charge is 50 C and the volume is 80 m3.

Solution:

We have,

q = 50

V = 80

Using the formula we have,

ρ = q/V

= 50/80

= 0.625 Cm-3

Problem 7: Calculate the charge if the volume charge density of a surface is 1 Cm-3 and volume is 25 m3.

Solution:

We have,

ρ = 1

V = 25

Using the formula we have,

ρ = q/V

=> q = ρV

= 1 (25)

= 25 C

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