Numerical on Continuous Charge Distribution Class 12

        In this article, we'll delve into practicing numerical on continuous charge distribution from NCERT Chapter 1 of Class 12 Physics. 

Continuous Charge Distribution

    There are three types of continuous distribution 

1.    Linear Charge Distribution (`\lamda = \frac{q}{l}`)

2.    Surface Charge Distribution (`\sigma = \frac{q}{A}`)

3.    Volume Charge Distribution  (`\rho = \frac{q}{V}` )

Numericals with solutions

    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}`

    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}`

    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}`

    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

    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
    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
    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`
    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}`

    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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Chapter  1:  ELECTRIC CHARGE AND FIELDS

PHYSICS NOTES

• Electric Charge, Basic Properties of Electric Charge, Conductors and Insulators, and Methods of Charging
• Continuous Charge Distribution
• Coulomb's Law, Electroscope, Properties of Charge, Quantization of Charge.
• Coulomb's Law in Vector Form
• Principle of Superposition of Charges
• Electric Field and Field Lines, Types of Electric Field, Electric Field Due to a Point Charge, Electric Filed Due to a System of Charges, Electric Field lines and their Properties
• Gauss's Law of Electrostatic, Definition of Gauss's Law, Formula, Electric Charge and its Four Properties, Applications of Garss's Law, Gauss's Law, and Important Points
• Prove of Gauss Theorem in Electrostatics
• Electric Field due to Conducting Sphere
• Electric Field due to Conducting Hollow Sphere
• Electric Field due to Non-Conducting Sphere
• Electric Field due to Infinitely Long Charged Wire
• Electric Field due to Infinite Conducting Sheet of Charge..
• Electric Field due to Non-Conducting Plane Sheet
• Electric Flux
• Definitions of Electric Dipole, Electric Field due to Electric Dipole on Axial Line..
• Electric Field Due to Electric Dipole on Equatorial Line
• Torque on a Dipole in Uniform Electric Field
• Work Done in Rotating a Dipole..
• Dielectric Constant