Field Due To Dipole And Continuous Charge Distributions - Continuous Charge Distribution
- In this lecture, we will discuss the topic of “Field Due To Dipole And Continuous Charge Distributions” with a focus on continuous charge distribution.
- We will explore the concept of electric field generated by continuous charge distributions.
- This topic is an important part of the Physics syllabus for 12th Boards examination.
- Understanding the field due to continuous charge distributions is crucial in various applications such as analyzing the behavior of conductors and insulators.
- Let’s begin by revisiting the concept of electric field and its significance.
Electric Field
- Electric field (E) is a vector quantity that describes the force experienced by a positive test charge (q) at a given point in space.
- It is defined as the force per unit positive charge at that point.
- Electric field (E) is given by the equation:
- E = F/q
- Where F is the force experienced by the test charge (q).
Electric Field Lines
- Electric field lines are used to visually represent the magnitude and direction of the electric field at different points in space.
- Field lines start from positive charges and terminate on negative charges.
- The density of field lines represents the strength of the electric field.
- Field lines do not intersect each other.
Electric Field Due to a Point Charge
- The electric field due to a point charge is given by Coulomb’s Law.
- For a point charge q, located at the origin, the electric field at a point P at distance r from the charge is given by:
- E = k * (q/r^2)
- Where k is the electrostatic constant.
- A uniformly charged ring is a circular ring with a constant charge per unit length (λ).
- The electric field at a point on the axis of the ring is given by:
- E = (k * Q * z) / ( (z^2 + R^2)^(3/2) )
- Where Q is the total charge on the ring and z is the distance of the point from the center of the ring.
- A uniformly charged disk is a flat circular disk with a constant charge per unit area (σ).
- The electric field at a point on the axis of the disk is given by:
- E = (k * σ * z) / ( 2 * ε_0 ) * ( √(1 - r^2/R^2) )
- Where σ is the surface charge density and z is the distance of the point from the center of the disk.
Electric Field Due to a Line Charge
- A line charge is an infinitely long line with a constant linear charge density (λ).
- The electric field at a point P, located at a perpendicular distance r from the line charge, is given by:
- E = (k * λ) / r
- Where λ is the linear charge density.
- A uniformly charged rod is a straight rod with a constant charge per unit length (λ).
- The electric field at a point P, located at a perpendicular distance r from the rod, is given by:
- E = (k * λ) / (2πε_0) * (1 / r) * (1 - cosθ)
- Where λ is the linear charge density and θ is the angle between the positive direction of the rod and the line joining the point P to the rod.
- A uniformly charged sphere is a solid sphere with a constant charge per unit volume (ρ).
- The electric field at a point outside the sphere is the same as that of a point charge at the center of the sphere.
- The electric field at a point inside the sphere is zero.
- A non-uniformly charged sphere is a solid sphere with a varying charge density.
- To determine the electric field at a point outside the sphere, we need to integrate the contributions from all infinitesimally small charge elements of the sphere.
- The electric field at a point inside the sphere can be calculated using Gauss’s Law.
- Continuous Charge Distribution - Electric Field
- A continuous charge distribution refers to a system in which charges are distributed continuously throughout a region.
- In such a distribution, we can consider an infinitesimally small charge element (∆q) and calculate the electric field due to that element.
- To determine the electric field at a given point, we need to integrate the contributions from all infinitesimally small charge elements in that region.
- The electric field due to a continuous charge distribution can be calculated using the principle of superposition.
- Let’s consider some examples of continuous charge distributions and calculate the electric field at different points.
- Electric Field Due to a Line Charge Distribution
- A line charge distribution is a distribution of charges along a line with a varying charge density (ρ).
- To calculate the electric field at a point due to a line charge distribution, we need to integrate the contributions from all infinitesimally small charge elements along the line.
- The electric field at a point P, located at a distance r from the line charge, is given by:
- E = ∫ (k * ρ * dl) / r
- Where k is the electrostatic constant, dl is an element of length along the line, and the integral is taken over the entire length of the line charge.
- Electric Field Due to a Surface Charge Distribution
- A surface charge distribution is a distribution of charges on a two-dimensional surface with a varying charge density (σ).
- To calculate the electric field at a point due to a surface charge distribution, we need to integrate the contributions from all infinitesimally small charge elements on the surface.
- The electric field at a point P, located at a distance r from the surface charge, is given by:
- E = ∫ (k * σ * dA) / r^2
- Where k is the electrostatic constant, dA is an element of area on the surface, and the integral is taken over the entire surface charge.
- Electric Field Due to a Volume Charge Distribution
- A volume charge distribution is a distribution of charges in a three-dimensional region with a varying charge density (ρ).
- To calculate the electric field at a point due to a volume charge distribution, we need to integrate the contributions from all infinitesimally small charge elements in the volume.
- The electric field at a point P, located at a distance r from the volume charge, is given by:
- E = ∫ (k * ρ * dV) / r^2
- Where k is the electrostatic constant, dV is an element of volume, and the integral is taken over the entire volume charge.
- Electric Field Due to a Charged Cylinder
- A charged cylinder is a cylinder with a uniform charge density (λ) along its length.
- To calculate the electric field at a point due to a charged cylinder, we need to integrate the contributions from all infinitesimally small charge elements along the cylinder.
- The electric field at a point P, located at a distance r from the charged cylinder, is given by:
- E = ∫ (k * λ * ds) / r
- Where k is the electrostatic constant, λ is the linear charge density, ds is an element of arc length along the cylinder, and the integral is taken over the entire length of the cylinder.
- Electric Field Due to a Charged Ring
- A charged ring is a ring with a uniform charge density (σ) along its circumference.
- To calculate the electric field at a point due to a charged ring, we need to integrate the contributions from all infinitesimally small charge elements along the ring.
- The electric field at a point P, located at a distance r from the charged ring, is given by:
- E = ∫ (k * σ * dl) / r^2
- Where k is the electrostatic constant, σ is the surface charge density, dl is an element of length along the ring, and the integral is taken over the entire circumference of the ring.
- Electric Field Due to a Charged Disc
- A charged disc is a flat disc with a uniform charge density (σ) on its surface.
- To calculate the electric field at a point due to a charged disc, we need to integrate the contributions from all infinitesimally small charge elements on the disc.
- The electric field at a point P, located at a distance r from the charged disc, is given by:
- E = ∫ (k * σ * dA) / (2 * ε_0) * ( √(1 - r^2/R^2) )
- Where k is the electrostatic constant, σ is the surface charge density, dA is an element of area on the disc, ε_0 is the vacuum permittivity, and the integral is taken over the entire surface of the disc.
- Electric Field Due to a Charged Sphere Shell
- A charged sphere shell is a hollow sphere with a uniform charge density (σ) on its surface.
- To calculate the electric field at a point outside the charged sphere shell, we can use the principle of superposition and consider the shell as a point charge at the center of the shell.
- The electric field at a point inside the charged sphere shell is zero, as the contributions from the inner and outer surfaces of the shell cancel out.
- The electric field at a point outside the charged sphere shell is the same as that of a point charge at the center of the shell.
- Summary - Continuous Charge Distribution
- In summary, the electric field due to a continuous charge distribution can be calculated by integrating the contributions from all infinitesimally small charge elements in that distribution.
- The electric field due to a line charge distribution, surface charge distribution, and volume charge distribution can be calculated using appropriate integrals and charge densities.
- The electric field due to a charged cylinder, charged ring, and charged disc can be calculated by integrating the contributions from all infinitesimally small charge elements in the respective geometries.
- The electric field due to a charged sphere shell is the same as that of a point charge at the center of the shell.
- Understanding the electric field due to continuous charge distributions is essential in analyzing the behavior of various systems and phenomena in electromagnetism.
"
Electric Field Due to a Charged Plate
- A charged plate is a flat plate with a uniform charge density (σ) on its surface.
- To calculate the electric field at a point due to a charged plate, we need to integrate the contributions from all infinitesimally small charge elements on the plate.
- The electric field at a point P, located at a distance r from the charged plate, is given by:
- E = (σ / 2 * ε_0)
- Where σ is the surface charge density and ε_0 is the vacuum permittivity.
- The electric field due to a charged plate is uniform and independent of the distance from the plate.
Example:
- Consider a uniformly charged plate with a surface charge density of σ = 10 μC/m^2.
- Calculate the electric field at a point located 5 cm away from the plate.
Solution:
- Using the formula for the electric field due to a charged plate, we have:
- E = (σ / 2 * ε_0) = (10 * 10^-6 C/m^2) / (2 * 8.85 * 10^-12 F/m)
- E = 5.64 * 10^4 N/C
- The electric field at the point is 5.64 * 10^4 N/C, directed away from the plate.
Electric Field Due to a Charged Cylinder Shell
- A charged cylinder shell is a hollow cylinder with a uniform charge density (λ) on its surface.
- To calculate the electric field at a point due to a charged cylinder shell, we need to integrate the contributions from all infinitesimally small charge elements on the shell.
- The electric field at a point P, located at a distance r from the charged cylinder shell, is given by:
- E = (k * λ * r) / (2 * ε_0 * √(R^2 + r^2))
- Where k is the electrostatic constant, λ is the linear charge density, R is the radius of the cylinder, and ε_0 is the vacuum permittivity.
- The electric field due to a charged cylinder shell is uniform along the axis of the cylinder and varies inversely with the distance from the cylinder.
Example:
- Consider a charged cylinder shell with a linear charge density of λ = 2 μC/m and a radius of R = 0.1 m.
- Calculate the electric field at a point located 0.2 m away from the axis of the cylinder.
Solution:
- Using the formula for the electric field due to a charged cylinder shell, we have:
- E = (k * λ * r) / (2 * ε_0 * √(R^2 + r^2))
= (9 * 10^9 N m^2/C^2 * 2 * 10^-6 C/m * 0.2 m) / (2 * 8.85 * 10^-12 F/m * √(0.1^2 + 0.2^2))
- E = 2.02 * 10^7 N/C
- The electric field at the point is 2.02 * 10^7 N/C, directed away from the axis of the cylinder.
Electric Field Due to a Charged Sphere
- A charged sphere is a solid sphere with a uniform charge density (ρ) throughout its volume.
- To calculate the electric field at a point due to a charged sphere, we need to integrate the contributions from all infinitesimally small charge elements within the sphere.
- The electric field at a point P, located at a distance r from the center of the charged sphere, is given by:
- E = (k * Q * r) / (4π * ε_0 * R^3)
- Where k is the electrostatic constant, Q is the total charge of the sphere, R is the radius of the sphere, and ε_0 is the vacuum permittivity.
Example:
- Consider a charged sphere with a total charge of Q = 4 μC and a radius of R = 0.2 m.
- Calculate the electric field at a point located 0.5 m away from the center of the sphere.
Solution:
- Using the formula for the electric field due to a charged sphere, we have:
- E = (k * Q * r) / (4π * ε_0 * R^3)
= (9 * 10^9 N m^2/C^2 * 4 * 10^-6 C * 0.5 m) / (4π * 8.85 * 10^-12 F/m * (0.2 m)^3)
- E = 3.88 * 10^6 N/C
- The electric field at the point is 3.88 * 10^6 N/C, directed away from the center of the sphere.
Electric Field Due to a Charged Solid Sphere
- A charged solid sphere is a solid sphere with a non-uniform charge density (ρ) throughout its volume.
- To calculate the electric field at a point due to a charged solid sphere, we need to integrate the contributions from all infinitesimally small charge elements within the sphere.
- Depending on the charge distribution, the electric field inside and outside the sphere can vary.
Example:
- Consider a charged solid sphere with a charge distribution given by ρ = k * r, where k is a constant and r is the distance from the center of the sphere.
- Calculate the electric field at a point located 0.3 m away from the center of the sphere.
Solution:
- Since the charge distribution is not given explicitly, we cannot directly calculate the electric field using a simple formula.
- We need to consider the contributions from all infinitesimally small charge elements within the sphere and integrate to find the total electric field at the point.
- The integration process may involve using spherical coordinates and the appropriate volume element, depending on the charge distribution.
Electric Field Due to an Infinite Line of Charge
- An infinite line of charge is a line with a constant linear charge density (λ) extending infinitely in both directions.
- To calculate the electric field at a point due to an infinite line of charge, we can use Gauss’s Law.
- Gauss’s Law states that the electric field through a closed surface is proportional to the net charge enclosed by that surface.
- The electric field at a point P, located at a distance r from the line of charge, is given by:
- E = (k * λ) / r
- Where k is the electrostatic constant and λ is the linear charge density.
Example:
- Consider an infinite line of charge with a linear charge density of λ = 5 nC/m.
- Calculate the electric field at a point located 10 cm away from the line of charge.
Solution:
- Using the formula for the electric field due to an infinite line of charge, we have:
- E = (k * λ) / r = (9 * 10^9 N m^2/C^2 * 5 * 10^-9 C/m) / 0.1 m
- E = 4.5 * 10^6 N/C
- The electric field at the point is 4.5 * 10^6 N/C, directed radially outward from the line of charge.
Electric Field Due to a Charged Wire
- A charged wire is a straight wire with a uniform charge density (λ) distributed along its length.
- To calculate the electric field at a point due to a charged wire, we need to integrate the contributions from all infinitesimally small charge elements along the wire.
- The electric field at a point P, located at a distance r from the wire, is given by:
- E = (k * λ) / (2πε_0) * (1 / r) * (1 - cosθ)
- Where k is the electrostatic constant, λ is the linear charge density,