Field Due To Dipole And Continuous Charge Distributions - Solid Angle

• Introduction to Field Due To Dipole And Continuous Charge Distributions
• Definition of Solid Angle
• Solid Angle Formula: Ω = A / r²
• Relation between Solid Angle and Steradian
• Calculation of Solid Angle
  • Example 1: Calculating the Solid Angle for a Point on the Surface of a Sphere
  • Example 2: Determining the Solid Angle Subtended by a Cone
• Solid Angle of a Point Charge
• Solid Angle of an Infinitely Long Wire
• Solid Angle of a Charged Disk
• Solid Angle of a Charged Sphere

11. Solid Angle of a Point Charge:

• For a point charge q, the solid angle subtended at any point in space is given by: Ω = q / (4πε₀r²)
• ε₀ is the permittivity of free space and r is the distance between the point charge and the observation point.
• Example: Given a point charge of +2μC, calculate the solid angle it subtends at a distance of 1 meter.

12. Solid Angle of an Infinitely Long Wire:

• Consider an infinitely long wire carrying a charge per unit length λ.
• The solid angle at a distance r from the wire is given by: Ω = λ / (2πε₀r)
• Example: Find the solid angle subtended by a uniformly charged wire with a charge density of 5 μC/m at a distance of 2 meters.

13. Solid Angle of a Charged Disk:

• The solid angle subtended by a charged disk of radius R and surface charge density σ at a point on its axis is given by: Ω = 2π(1 - cosθ)
• θ is the angle between the axis of the disk and the line joining the observation point to the center of the disk.
• Example: Calculate the solid angle subtended at a point on the axis of a uniformly charged disk with radius 10 cm and surface charge density 3 μC/m².

14. Solid Angle of a Charged Sphere:

• For a uniformly charged sphere of radius R and total charge Q, the solid angle subtended at a point inside the sphere is given by: Ω = 4π(1 - cosθ)
• θ is the angle between the axis passing through the point inside the sphere and the line joining the observation point to the center of the sphere.
• Example: Determine the solid angle subtended at a point inside a charged sphere of radius 5 cm and total charge 12 μC.

15. Calculation of Solid Angle:

• Solid angle can be calculated by dividing the subtended area by the square of the distance from the observation point.
• Example: Find the solid angle subtended by a hemisphere of radius 6 cm at a distance of 4 cm from its center.

16. Example 1: Calculating the Solid Angle for a Point on the Surface of a Sphere:

• Consider a point P on the surface of a sphere with radius r.
• The solid angle subtended by an area element dA at point P is given by: dΩ = dA / r²
• Example: Calculate the solid angle subtended by a small area element dA on the surface of a sphere of radius 8 cm.

17. Example 2: Determining the Solid Angle Subtended by a Cone:

• Consider a cone with vertex at point O and angle α.
• The solid angle subtended by the cone at a point P at distance r from O is given by: Ω = 2π(1 - cos(α/2))
• Example: Find the solid angle subtended by a cone with angle 60 degrees at a point 10 cm away from its vertex.

18. Relation between Solid Angle and Steradian:

• Solid angle is the three-dimensional analog of plane angle.
• The SI unit of solid angle is the steradian (sr).
• 1 steradian is equal to the solid angle subtended by a spherical surface area equal to the square of the sphere’s radius.
• Relationship: 1 steradian = 1 square radian = 1 radian².

19. Applications of Solid Angle in Physics:

• Solid angle is commonly used in various areas of physics, including:
  • Calculations involving electric field due to charge distributions
  • Calculation of radiant intensity and luminous flux in optics
  • Calculating the intensity distribution of radiation from a point source
  • Measurement and analysis of radiation in nuclear physics

20. Summary:

• Solid angle is a measure of the size of a region in three-dimensional space.
• It is defined as the ratio of the area subtended by a region on a sphere to the square of the radius of the sphere.
• Solid angle is expressed in steradians (sr) and is used in various calculations in physics.
• It finds applications in electric field calculations, optics, and nuclear physics.
• Understanding solid angle is essential for analyzing and predicting the behavior of electromagnetic and radiation phenomena.

21. Electric Field Due to an Electric Dipole:

• An electric dipole consists of two equal and opposite charges separated by a distance d.
• The electric field due to an electric dipole at a point P in the axial region (far away from the dipole) is given by: E = (k * p) / r² where k is the electrostatic constant, p is the electric dipole moment, and r is the distance from the dipole.
• The direction of the electric field is away from the positive charge and towards the negative charge.

22. Electric Field Due to a Continuous Charge Distribution:

• For a continuous charge distribution, we need to integrate the contribution of all the individual charge elements to find the total electric field at a point.
• Electric field due to a ring of charge: E = (k * dq * sinθ) / (4πε₀r²) where dq is the charge element, θ is the angle between the line joining the charge element to the observation point, and r is the distance from the charge element.

23. Electric Field Due to a Uniformly Charged Ring:

• A uniformly charged ring with radius R and charge Q produces an electric field at its center given by: E = (k * Q) / (2πε₀R³)
• Example: Calculate the electric field produced at the center of a uniformly charged ring with a radius of 10 cm and a charge of 4 μC.

24. Electric Field Due to a Uniformly Charged Disk:

• A uniformly charged disk with radius R and surface charge density σ produces an electric field at a point on its axis given by: E = (σ / (2ε₀)) * (1 - (z / √(R² + z²)))
• Example: Find the electric field at a point on the axis of a uniformly charged disk with a radius of 8 cm and a surface charge density of 10 μC/m². The point is located at a distance of 4 cm from the disk.

25. Electric Field Due to a Charged Sphere:

• A uniformly charged sphere with radius R and total charge Q produces an electric field at a point outside the sphere given by: E = (k * Q) / (4πε₀r²)
• Example: Determine the electric field produced by a uniformly charged sphere with a radius of 6 cm and a total charge of 2 μC at a point 10 cm away from its center.

26. Electric Field Due to an Infinite Line of Charge:

• An infinite line of charge with charge per unit length λ produces an electric field at a point a distance r from the line given by: E = (λ / (2πε₀r))
• Example: Calculate the electric field produced by an infinite line of charge with a charge density of 8 μC/m at a distance of 5 cm from the line.

27. Electric Field Due to an Infinitely Long Charged Cylinder:

• An infinitely long charged cylinder with linear charge density λ produces an electric field at a point outside the cylinder given by: E = (λ / (2πε₀r))
• Example: Find the electric field produced by an infinitely long charged cylinder with a linear charge density of 6 μC/m at a distance of 3 cm from the cylinder.

28. Electric Field Due to a Charged Rod:

• A charged rod with length L and charge Q produces an electric field at a point on its axis a distance x away from one end of the rod given by: E = (k * Q * x) / (4πε₀ (x² + L²/4)^(3/2))
• Example: Determine the electric field produced by a charged rod with a length of 4 cm and a charge of 1 μC at a point on its axis located 2 cm away from one end.

29. Electric Field Due to a Line of Charges:

• A line of charges with charge density ρ produces an electric field at a point P a distance r away given by: E = (k * λ) / (2πε₀r) where λ is the linear charge density of the line of charges.
• Example: Calculate the electric field produced by a line of charges with a linear charge density of 3 μC/m at a distance of 8 cm from the line.

30. Summary:

• Field due to dipole and continuous charge distributions can be calculated using the concept of solid angle and integral calculus.
• For electric dipole, the electric field is inversely proportional to the square of the distance from the dipole, and its direction is dependent on the orientation of the dipole.
• Continuous charge distributions, such as rings, disks, spheres, infinite lines, and charged cylinders, have specific formulas for calculating the electric field at different observation points.
• Examples have been provided to illustrate the application of these formulas.
• Understanding the electric field due to dipole and continuous charge distributions is crucial in various areas of physics, including electromagnetism and electrostatics.