What is Charge Density?
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
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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
FAQs on Charge Density Formula
1. What is charge density and what are its main types in Physics?
Charge density is a measure of the concentration of electric charge in a given space. It quantifies how much electric charge is accumulated per unit dimension. Depending on the dimension over which the charge is distributed, it is classified into three main types:
- Linear Charge Density (λ): For charge distributed along a one-dimensional line or a thin wire.
- Surface Charge Density (σ): For charge spread over a two-dimensional surface or a thin sheet.
- Volume Charge Density (ρ): For charge distributed throughout a three-dimensional volume.
2. What is the formula and SI unit for linear charge density?
The formula for linear charge density (λ) is the total charge (q) distributed uniformly over a given length (l).
Formula: λ = q / l
The SI unit for linear charge density is coulombs per meter (C/m).
3. How do you calculate surface charge density, and what is its SI unit?
You can calculate surface charge density (σ) by dividing the total charge (q) by the total surface area (A) over which it is spread.
Formula: σ = q / A
The SI unit for surface charge density is coulombs per square meter (C/m²).
4. What is the formula for volume charge density, and what is its SI unit?
The formula for volume charge density (ρ) is the total charge (q) distributed throughout a specific volume (V).
Formula: ρ = q / V
The SI unit for volume charge density is coulombs per cubic meter (C/m³).
5. Can charge density be negative, and what does that signify?
Yes, charge density can be negative. The sign of the charge density directly corresponds to the sign of the electric charge being distributed. A positive charge density indicates a distribution of positive charges (like protons or positive ions). Conversely, a negative charge density signifies a distribution of negative charges (like electrons), creating an electric field that points towards the charge distribution.
6. How is charge density used to calculate the electric field of a continuous charge distribution?
For continuous charge distributions (like a charged rod or sphere), we cannot use the simple point-charge formula. Instead, charge density is essential for applying calculus. It allows us to define a small charge element (dq) in terms of a small geometric element (dl, dA, or dV). By integrating the electric field contributions from all such charge elements over the entire distribution, we can find the total electric field. This concept is fundamental to applying Gauss's Law, which relates the electric flux through a closed surface to the total charge enclosed, determined via charge density.
7. How is charge density (ρ) in electrostatics different from current density (J)?
While both concepts relate to electric charge, they describe different physical phenomena:
- Charge Density (ρ, σ, λ): Describes the amount of static (non-moving) charge per unit volume, area, or length. It is a scalar quantity for volume charge density.
- Current Density (J): Describes the rate of flow of charge (electric current) per unit cross-sectional area. It is a vector quantity, as it has both magnitude and direction, indicating the direction of charge flow.
In simple terms, charge density is about how much charge is 'sitting' in a space, while current density is about how much charge is 'moving' through a space.
8. In what real-world applications is the concept of charge density important?
The concept of charge density is crucial in many areas of science and engineering. Key applications include:
- Electromagnetism: It serves as a primary source term in Maxwell's equations, which form the foundation of classical electromagnetism.
- Capacitors: The ability of a capacitor to store energy depends on the surface charge density that builds up on its plates.
- Semiconductor Physics: Understanding charge density is vital for designing transistors and other electronic components.
- Chemistry: The charge density around an atom or molecule influences its chemical reactivity, bonding properties, and interactions with other molecules.
