Field Due To Dipole And Continuous Charge Distributions - Electrostatics

  • In this lecture, we will discuss the concept of field due to a dipole and continuous charge distributions.
  • Field is a physical quantity that fills the space around a charged object.
  • It represents the influence that a charged object exerts on other charged objects or on a test charge.
  • The field at any point in space is a vector quantity and is given by the electric field vector E.
  • The field due to a dipole or continuous charge distributions can be calculated using specific formulas.

Field Due To an Electric Dipole

  • An electric dipole consists of two equal and opposite charges, separated by a distance d.
  • The dipole moment (p) of an electric dipole is given by the product of the charge (q) and the separation distance (d).
  • The electric field at any point on the axial line of a dipole is given by the formula: E = (kp) / (r^2) where k is the electrostatic constant and r is the distance from the dipole.
  • The electric field at any point on the equatorial line of a dipole is given by the formula: E = (kp) / (r^3)
  • The direction of the electric field at different points is determined by the dipole orientation.

Field Due To Continuous Charge Distributions

  • In some cases, the charge is not concentrated at specific points but is distributed continuously along a line, surface, or volume.
  • The electric field due to a continuous charge distribution can be calculated using integration.
  • For a line charge distribution with linear charge density (λ), the electric field at a distance (r) from the line is given by the formula: E = (2kλ) / (r)
  • For a surface charge distribution with surface charge density (σ), the electric field at a distance (r) from the surface is given by the formula: E = (σ) / (2ε₀)
  • For a volume charge distribution with volume charge density (ρ), the electric field at a distance (r) from the volume is given by the formula: E = (ρ) / (3ε₀)
  • Here, ε₀ is the permittivity of free space.

Electric Field Lines

  • Electric field lines are imaginary lines used to represent the electric field.
  • Electric field lines originate from positive charges and terminate on negative charges.
  • The direction of the electric field at any point is tangent to the electric field line passing through that point.
  • The density of electric field lines represents the strength of the electric field.
  • Electric field lines never intersect each other.
  • Electric field lines are closer together where the electric field is stronger and spread out where the electric field is weaker.

Electric Field Intensity

  • The electric field intensity (E) at any point is defined as the force experienced by a unit positive charge placed at that point.
  • It is a measure of the strength of the electric field at that point.
  • Electric field intensity is a vector quantity and its direction is the same as the electric field vector at that point.
  • The electric field intensity at a point due to multiple charges is the vector sum of the electric field intensities due to each individual charge. Let’s now solve some numerical examples to understand these concepts better.
  1. Field Due To Dipole (Axial Line)
  • Electric field at any point on the axial line of a dipole: E = (kp) / (r^2)
  • The direction of the field is determined by the orientation of the dipole.
  • The field points from the positive charge towards the negative charge.
  • The field is stronger closer to the dipole and decreases as the distance increases.
  • Examples: Field due to Earth’s magnetic dipole, field between two opposite magnetic poles.
  1. Field Due To Dipole (Equatorial Line)
  • Electric field at any point on the equatorial line of a dipole: E = (kp) / (r^3)
  • The direction of the field at each point is perpendicular to the line joining the charges.
  • The field is strongest at the midpoint between the charges.
  • The field is zero at the center of the dipole.
  • Examples: Electric field at the center of a bar magnet, field at the center of an electric dipole.
  1. Field Due To Line Charge Distribution
  • Electric field due to line charge distribution: E = (2kλ) / (r)
  • The field is directed radially outwards or inwards, depending on the sign of the charge.
  • The field is strongest closer to the line and decreases with distance.
  • Example: Electric field around an infinitely long uniformly charged wire.
  1. Field Due To Surface Charge Distribution
  • Electric field due to surface charge distribution: E = (σ) / (2ε₀)
  • The field is perpendicular to the surface at each point.
  • The field is stronger closer to the surface and decreases with distance.
  • Example: Electric field near the surface of a uniformly charged conductor.
  1. Field Due To Volume Charge Distribution
  • Electric field due to volume charge distribution: E = (ρ) / (3ε₀)
  • The direction of the field is determined by the charge distribution.
  • The field is stronger closer to the volume and decreases with distance.
  • Example: Electric field inside a uniformly charged sphere.
  1. Importance of Electric Field Lines
  • Electric field lines provide a visual representation of the electric field.
  • They help understand the direction and strength of the electric field at different points.
  • Electric field lines provide insights into the behavior of charges in the presence of electric fields.
  • They aid in solving electrostatic problems and predicting charge movements.
  • Electric field lines simplify the visualization of complex charge distributions.
  1. Characteristics of Electric Field Lines
  • Electric field lines originate from positive charges and terminate on negative charges.
  • They never intersect each other.
  • The number of field lines per unit area represents the magnitude of the electric field.
  • Field lines are closer together where the field is stronger and further apart where the field is weaker.
  • Field lines are always tangent to the electric field vectors at each point.
  1. Electric Field Intensity
  • Electric field intensity (E) measures the strength of the electric field at a point.
  • It is the force experienced by a unit positive charge placed at that point.
  • Electric field intensity is a vector quantity with magnitude and direction.
  • The direction of E is the same as the direction of the electric field at that point.
  • The electric field intensity due to multiple charges is the vector sum of the intensities due to each charge.
  1. Electric Field Intensity Calculation
  • To calculate the electric field intensity, we consider the contribution of individual charges.
  • Sum the electric field intensities due to each charge vectorially to find the total field intensity.
  • The magnitude of the net electric field intensity is determined by the superposition principle.
  • Electric field intensity can vary in different regions depending on the distribution of charges.
  • Examples: Calculating net electric field intensity due to multiple point charges or charge distributions.
  1. Summary
  • The electric field vector represents the influence of a charged object on other charges or test charges.
  • Electric field due to a dipole is different on the axial line and the equatorial line.
  • Line, surface, and volume charge distributions have specific formulas for electric field calculation.
  • Electric field lines provide insights into the direction and strength of the electric field.
  • Electric field intensity measures the strength of the electric field at a point and is determined by the superposition principle.
  1. Electric Field Due To Dipole - Axial Line
  • Electric field at any point on the axial line: E = (kp) / (r^2)
  • The direction of the field is determined by the dipole orientation.
  • Field points from the positive charge towards the negative charge.
  • Field is stronger closer to the dipole and decreases with distance.
  • Examples: Field due to Earth’s magnetic dipole, field between two opposite magnetic poles.
  1. Electric Field Due To Dipole - Equatorial Line
  • Electric field at any point on the equatorial line: E = (kp) / (r^3)
  • The direction of the field is perpendicular to the dipole axis.
  • Field is strongest at the midpoint between the charges.
  • Field is zero at the center of the dipole.
  • Examples: Electric field at the center of a bar magnet, field at the center of an electric dipole.
  1. Electric Field Due To Line Charge Distribution
  • Electric field due to line charge distribution: E = (2kλ) / (r)
  • Field is directed radially outwards or inwards, depending on charge sign.
  • Field is strongest closer to the line and decreases with distance.
  • Example: Electric field around an infinitely long uniformly charged wire.
  1. Electric Field Due To Surface Charge Distribution
  • Electric field due to surface charge distribution: E = (σ) / (2ε₀)
  • Field is perpendicular to the surface at each point.
  • Field is stronger closer to the surface and decreases with distance.
  • Example: Electric field near the surface of a uniformly charged conductor.
  1. Electric Field Due To Volume Charge Distribution
  • Electric field due to volume charge distribution: E = (ρ) / (3ε₀)
  • Field direction is determined by charge distribution.
  • Field is stronger closer to the volume and decreases with distance.
  • Example: Electric field inside a uniformly charged sphere.
  1. Importance of Electric Field Lines
  • Electric field lines visually represent the electric field.
  • They indicate the direction and strength of the electric field.
  • Electric field lines aid in understanding charge behavior.
  • They assist in solving electrostatic problems.
  • Electric field lines simplify complex charge distributions.
  1. Characteristics of Electric Field Lines
  • Electric field lines originate from positive charges and terminate on negative charges.
  • Field lines never intersect each other.
  • Density of field lines represents the electric field strength.
  • Lines are closer together where the field is stronger.
  • Field lines are tangent to the electric field vectors at each point.
  1. Electric Field Intensity
  • Electric field intensity (E) measures the field strength.
  • It is the force experienced by a unit positive charge.
  • E is a vector quantity with magnitude and direction.
  • E is parallel to the electric field at a point.
  • Net E is the vector sum of individual E’s.
  1. Electric Field Intensity Calculation
  • Calculate E by considering individual charge contributions.
  • Vectorially sum the E’s to find the total field intensity.
  • Net E magnitude depends on superposition principle.
  • Electric field intensity can vary in different regions.
  • Examples: Calculating net E due to multiple charges.
  1. Summary
  • The electric field represents the influence of a charged object on other charges or test charges.
  • Field due to a dipole has specific formulas on the axial and equatorial lines.
  • Line, surface, and volume charge distributions have different field formulas.
  • Electric field lines aid in understanding field direction and strength.
  • Electric field intensity measures the strength of the electric field and is determined by superposition principle.