This article will delve into practicing numerical continuous charge distribution from NCERT Chapter 1 of Class 12 Physics.
Continuous Charge Distribution - Definition:
If the charge given to an object spreads uniformly along its length, over its surface, or throughout its entire volume, it is called a continuous charge distribution.
Discrete Charge distribution:
When charge is present in the form of individual, separate charged particles, it is called a discrete charge distribution, and it is also known as a non-uniform charge distribution.
- Each charge can be counted separately.
- Charge varies from point to point.
- Charge density is different at different positions.
- Example: A system of point charges like, ` q_1, q_2, q_3....`
Continuous Charge distribution:
When charge is spread continuously over a length, surface, or volume, it is called a continuous charge distribution.
- Charge is evenly spread.
- Charge density is constant at every point.
- Charges are very closely packed and cannot be treated individually.
- This type of charge distribution is described using charge density.
There are three types of continuous distributions
1. Linear Charge Distribution (`\lamda = \frac{q}{l}`):
If the carge given to a conductor is uniformly distributed along its entire length, it is called linear charge distribution.
The amount of charge per unit length is called linear charge density, and it is represented by `\lamda`(lambda).
Linear Charge Density `\lamda = \frac{Q}{L}`
Where:
`\lamda =` Charge per unit length (C/m)
Q = Total charge (C)
L = Length (m)
2. Surface Charge Distribution (`\sigma = \frac{q}{A}`):
If the charge given to a conductor is uniformly distributed over its entire surface, it is called surface charge distribution.
The amount of charge per unit area is called surface charge density, and it is represented by `\sigma` (sigma).
Surface Charge Density `\sigma = \frac{Q}{L}`
Where:
`\sigma =` Charge per unit area (`C/m^2`)
Q = Total charge (C)
A = Surface area (`m^2`)
3. Volume Charge Distribution (`\rho = \frac{q}{V}` ):
If the charge given to a conductor is uniformly distributed throughout its entire volume, it is called volume charge distribution.
The amount of charge per unit volume is called volume charge density, and it is represented by `\rho` (rho).
Linear Charge Density `\rho = \frac{Q}{L}`
Where:
`\rho =` Charge per unit length (`C/m^3`)
Q = Total charge (C)
V = Volume (`m^3`)
Important Points:
- In conductors, charge always stays on the outer surface.
- At sharp edges, charge density ishigher.
- In insulators, charge may remain fixed and not redistribute easily.
- Charge distribution helps in calculating the electric field and potential.
Numericals with solutions
Q1. A wire of length 2 meters has a total charge of 4 Coulombs uniformly distributed on it. What is the linear charge density?
We can use this formula for linear charge density
`\lambda = \frac{Q}{L}`
Where,
`\lamda =` linear charge density
Q = Total Charge, and
L = Length of wire
According to question
Q = 4 Coulombs
L = 2 meters
Then,
`\lambda = \frac{Q}{L}`
`\lambda = \frac{4}{2}\frac{C}{m}`
`\lambda = 2 \frac{C}{m}`
So, the linear charge density is 2 `\frac{C}{m}`
Q2. A surface with an area of 0.5 square meters has a total charge of 8 Coulombs uniformly distributed on it. What is the surface charge density?
We can use the formula for surface charge density
`\sigma = \frac{Q}{A}`
Where,
`\sigma = ` Surface charge density,
Q = Total Charge, and
A = Area
According to Question
Q = 8 Coulombs
A = 0.5 `m^2`
Then,
`\sigma = \frac{Q}{A}`
`\sigma = \frac{8}{0.5}\frac{C}{m^2}`
`\sigma = \frac{80}{5}\frac{C}{m^2}`
`\sigma = 16 \frac{C}{m^2}`
Q3. A cube with a volume of 10 cubic meters has a total charge of 20 Coulombs uniformly distributed within it. What is the volume charge density?
Solution
We can use the formula for volume charge density
`\sigma = \frac{Q}{V}`
Where,
`\rho = ` Volume charge density,
Q = Total Charge, and
V = Volume
According to Question
Q = 20 Coulombs
A = 10 `m^3`
Then,
`\sigma = \frac{Q}{V}`
`\rho = \frac{20}{10}\frac{C}{m^3}`
`\rho = 2 \frac{C}{m^3}`
Q4. If the linear charge density on a wire is 0.1 Coulomb/meter and the length of the wire is 5 meters, what is the total charge on the wire?
Solution
We can use this formula for linear charge density
`\lambda = \frac{Q}{L}`
`Q = \lambda \times L`
Where,
`\lamda =` linear charge density
Q = Total Charge, and
L = Length of wire
According to question
`\lambda` = 0.1 Coulombs/meter
L = 5 meters
Then,
`Q = 0.1 \times 5`
`Q = 0.5` Coulombs
Q5. The surface charge density on a circular plate is 2 Coulombs/square meter. If the area of the plate is 3 square meters, what is the total charge on the plate?
Solution
We can use this formula for surface charge density
`\sigma = \frac{Q}{A}`
`Q = \sigma \times A`
Where,
`\sigma =` Surface charge density
Q = Total Charge, and
A = Area
According to question
`\sigma` = 2 Coulombs/mete`r^2`
`A = 3 m^2`
Then,
`Q = \sigma \times A`
`Q = 2 \times 3`
`Q = 6` Coulombs
We can use this formula for surface charge density
`\rho = \frac{Q}{V}`
`Q = \rho \times V`
Where,
`\rho =` Volume charge density
Q = Total Charge, and
V = Volume
According to question
`\rho` = 0.05 Coulombs/mete`r^3`
`V = 8 m^3`
Then,
`Q = 0.05 \times 8`
`Q = 0.4` Coulombs
Q6. A wire of length 3 meters has a linear charge density of 0.2 Coulomb/meter. What is the charge on a segment of the wire with a length of 1.5 meters?
Solution
We can use this formula for linear charge density
`\lambda = \frac{Q}{L}`
`Q = \lambda \times L`
Where,
`\lamda =` linear charge density
Q = Total Charge, and
L = Length of wire
According to question
`\lambda` = 0.2 Coulombs/meter
L = 1.5 meters
Then,
`Q = 0.2 \times 1.5`
`Q = 0.3` Coulombs
Q7.A surface has a surface charge density of 5 Coulombs/square meter. If the total charge on the surface is 15 Coulombs, what is the area of the surface?
Solution
We can use this formula for surface charge density
`\sigma = \frac{Q}{A}`
`A = \frac{Q}{\sigma}`
Where,
`\sigma =` Surface charge density
Q = Total Charge, and
A = Area
According to question
`\sigma` = 5 Coulombs/mete`r^2`
`Q = 15 Coulombs`
Then,
`A = \frac{Q}{\sigma}`
`A = \frac{Q15}{3}`
`A = 3 m^2`
Q8.The volume charge density within a cube is 0.01 Coulomb/cubic meter. If the volume of the cube is 100 cubic meters, what is the total charge within the cube?
We can use this formula for surface charge density
`\rho = \frac{Q}{V}`
`Q = \rho \times V`
Where,
`\rho =` Volume charge density
Q = Total Charge, and
V = Volume
According to question
`\rho` = 0.01 Coulombs/mete`r^3`
`V = 100 m^3`
Then,
`Q = 0.01 \times 100`
`Q = 1` Coulombs
Solution
We can use this formula for linear charge density
`\lambda = \frac{Q}{L}`
`L = \frac{Q}{\lambda}`
Where,
`\lamda =` linear charge density
Q = Total Charge, and
L = Length of wire
According to question
`\lambda` = 0.05 Coulombs/meter
Q = 1.5 Coulombs
Then,
`L = \frac{Q}{\lambda}`
`L = \frac{1.5}{0.05}`
`L = 30` meter
Solution
So, the surface charge density is 16 `\frac{C}{m^2}`
So, the volume charge density is 2 `\frac{C}{m^3}`
Q9. If the volume charge density within a sphere is 0.05 Coulomb/cubic meter and the volume of the sphere is 8 cubic meters, what is the total charge within the sphere?
Solution
If the linear charge density on a wire is 0.05 Coulomb/meter and the total charge on the wire is 1.5 Coulombs, what is the length of the wire?
Related Questions
1. What is the continuous charge distribution?
2. What is the difference between continuous and discrete charge distribution
3. What does charge distribution mean?
4. What are the three types of continuous charge distribution?
5. What are the dimensions of linear charge density?
6. What are the dimensions of area charge density?
7. What are the dimensions of volume charge density?
8. What are the units of linear charge density?
9. What are the units of area charge density?
10. What are the units of volume charge density?
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