Field Due To Dipole And Continuous Charge Distributions - Field Due To Dipole And Continuous Charge Distributions

Field Due To Dipole And Continuous Charge Distributions - Field Due To Dipole And Continuous Charge Distributions – An introduction

In this lecture, we will discuss the concept of the electric field due to dipole and continuous charge distributions.
We will understand the definition of a dipole and how it creates an electric field.
We will also explore the concept of continuous charge distributions and their effect on the electric field.
Lastly, we will analyze the relationship between the distance from a charge distribution and the electric field intensity.

Electric Field due to a Dipole

A dipole consists of two equal and opposite charges separated by a small distance.
The electric field at any point on the axial line of a dipole is given by the equation:
- E = 2kqd / R^3
Here, k is the Coulomb constant, q is the magnitude of the charge, d is the separation between the charges, and R is the distance from the dipole to the point where the electric field is to be calculated.

Electric Field due to an Electric Dipole

Consider an electric dipole with charges +q and -q separated by distance d.
When an external charge is brought near the dipole, it experiences a force due to the electric field produced by the dipole.
The electric field due to the dipole at a point P on the axial line of the dipole is given by:
- E = 2kqd / R^3
Here, R is the distance between the center of the dipole and point P.

Electric Field due to a Continuous Charge Distribution

In many cases, instead of a point charge, we deal with charge distributions that have continuous charge distribution.
For such distributions, we use calculus to sum up the contributions of infinitesimally small charge elements.
The electric field due to a continuous charge distribution at a point P is given by:
- E = k ∫ (dq / r^2)
Here, dq represents an infinitesimally small charge element, r is the distance between dq and point P, and the integration is done over the entire distribution.

Electric Field due to a Line Charge

Consider a line charge with a linear charge density λ.
The electric field due to a uniformly charged line at a point P at a perpendicular distance r from the line is given by:
- E = k λ / r
Here, λ is the charge per unit length along the line charge.

Electric Field due to a Ring of Charge

Consider a ring of charge with a radius R and total charge Q.
The electric field due to a uniformly charged ring at a point P on the axis of the ring is given by:
- E = kQz / (4πε₀ (R^2+z^2)^(3/2))
Here, z is the perpendicular distance between the center of the ring and point P, ε₀ is the permittivity of free space, and Q is the total charge on the ring.

Electric Field due to a Disk of Charge

Consider a disk of charge with radius R and total charge Q.
The electric field due to a uniformly charged disk at a point P on its axis is given by:
- E = (σ / 2ε₀)[1 - (z / √(R^2+z^2))]
Here, σ is the surface charge density of the disk and z is the perpendicular distance between the center of the disk and point P.

Electric Field due to a Spherical Shell of Charge

Consider a spherical shell of charge with radius R and total charge Q.
The electric field due to a uniformly charged spherical shell at a point P outside the shell is given by:
- E = kQ / R^2
Here, Q is the total charge on the shell and R is the radius of the shell.

Electric Field due to a Solid Sphere of Charge

Consider a solid sphere of charge with radius R and total charge Q.
The electric field due to a uniformly charged solid sphere at a point P outside the sphere is given by:
- E = kQ / R^2
Here, Q is the total charge on the sphere and R is the radius of the sphere.

Electric Field due to an Electric Dipole - Calculation

To calculate the electric field due to an electric dipole at a point P, we consider a coordinate system with the origin at the center of the dipole.
The electric field due to the positive charge at point P is given by:
- E+ = kq / (r - d/2)^2
The electric field due to the negative charge at point P is given by:
- E- = kq / (r + d/2)^2
The net electric field at point P is the vector sum of the individual fields, given by:
- E_net = E+ - E-

Electric Field due to an Electric Dipole - Calculation (contd.)

The net electric field at point P is the vector sum of the individual fields, given by:
- E_net = E+ - E-
This can be simplified to:
- E_net = kq (2d / (r^2 - (d/2)^2))
Here, r is the distance from the center of the dipole to point P.
It is important to note that the electric field due to a dipole decreases as the distance from the dipole increases.

Electric Field due to a Continuous Charge Distribution

To calculate the electric field due to a continuous charge distribution, we divide the distribution into infinitesimally small charge elements.
For each charge element, we calculate the electric field produced at the point P using Coulomb’s law:
- dE = k (dq / r^2)
Here, dq represents an infinitesimally small charge element, r is the distance between dq and point P.
To obtain the total electric field, we integrate the electric field contributions from all charge elements over the entire distribution.

Electric Field due to a Uniformly Charged Rod

Consider a rod of length L and total charge Q with a uniform charge distribution.
The electric field due to the rod at a point P on its axis is given by:
- E = (kQ / L) [(L/2 + x) / (√((L/2 + x)^2 + y^2)) - (L/2 - x) / (√((L/2 - x)^2 + y^2))]
Here, x is the distance from the center of the rod to point P along the axis, and y is the perpendicular distance from the rod to point P.

Electric Field due to a Uniformly Charged Plane Sheet

Consider a plane sheet of charge with surface charge density σ.
The electric field due to the plane sheet at a point P above or below the sheet is given by:
- E = (σ / 2ε₀)
Here, ε₀ is the permittivity of free space and σ is the surface charge density of the plane sheet.

Electric Field due to a Point Charge

The electric field due to a point charge at a point P in space is given by Coulomb’s law:
- E = kq / r^2
Here, q is the magnitude of the charge, r is the distance between the charge and point P, and k is the Coulomb constant.

Electric Field due to a Circular Arc of Charge

Consider a circular arc of charge with angle θ and total charge Q.
The electric field due to the circular arc at a point P on the axis of the arc is given by:
- E = (kQ / (R^2 + x^2)) (sin(θ/2))/(√((1+cos(θ/2))^2 + x^2))
Here, R is the radius of the arc, x is the distance from the center of the arc to point P along the axis, and θ is the angle subtended by the arc at the center of the circle.

Electric Field Intensity

Electric field intensity refers to the strength of the electric field at a given point in space.
The magnitude of the electric field intensity is given by the equation:
- E = F / q
Here, E represents the electric field intensity, F is the force experienced by a positive test charge placed at that point, and q is the magnitude of the test charge.
Electric field intensity is a vector quantity, with both magnitude and direction.

Electric Field Lines

Electric field lines are an important graphical representation of electric fields.
Electric field lines always 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 at that point.
The density of electric field lines represents the magnitude of the electric field - more lines in a given area indicates a stronger electric field.

Properties of Electric Field Lines

Electric field lines never cross each other. If they were to cross, there would be two possible directions for the electric field at that point, which violates the uniqueness of the electric field.
Electric field lines are perpendicular to the surface of a conductor at every point. This implies that the electric field inside a conductor is zero.
Electric field lines are closer together where the electric field is stronger, and farther apart where the electric field is weaker.
Electric field lines make a smooth continuous curve, with no sharp corners or ends. They extend to infinity or terminate on charges.

Electric Field due to a Uniformly Charged Sphere

Consider a uniformly charged sphere of radius R and total charge Q.
The electric field due to the uniformly charged sphere at a point P outside the sphere is given by:
- E = kQ / R^2
Here, Q is the total charge on the sphere and R is the radius of the sphere.

Electric Field due to a Thin Spherical Shell of Charge

Consider a thin spherical shell of charge with radius R and total charge Q.
The electric field due to the thin spherical shell at a point P outside the shell is given by:
- E = kQ / R^2
Here, Q is the total charge on the shell and R is the radius of the shell.

Superposition Principle for Electric Fields

The principle of superposition states that the total electric field at a point due to a system of charges is the vector sum of the individual electric fields at that point due to each charge.
Mathematically, it can be expressed as:
- E_total = E_1 + E_2 + E_3 + …
Here, E_total is the total electric field, E_1, E_2, E_3, … are the electric fields due to individual charges.

Electric Field due to a Charged Disk

Consider a disk of total charge Q and radius R.
The electric field due to the charged disk along its axis at a point P is given by:
- For P outside the disk: E = (σ / 2ε₀) (1 / √(1 + (z / R)^2))^3
- For P on the axis beyond the disk: E = (σR / 2ε₀z^2)
- Here, σ is the surface charge density of the disk and z is the perpendicular distance from the center of the disk to point P.

Electric Field due to Multiple Point Charges

To calculate the electric field at a point P due to multiple point charges, we can use the principle of superposition.
The total electric field at point P is the vector sum of the electric fields due to each individual charge at that point.
Mathematically, it can be expressed as:
- E_total = E_1 + E_2 + E_3 + …
Here, E_total is the total electric field, E_1, E_2, E_3, … are the electric fields due to individual charges.

Electric Field due to a Solid Cylinder of Charge

Consider a solid cylinder of charge with radius R, length L, and total charge Q.
The electric field due to the solid cylinder at a point P outside the cylinder is given by:
- E = kQ / (L + R)
Here, Q is the total charge on the cylinder, L is the length of the cylinder, and R is the radius of the cylinder.

Electric Field due to Two Point Charges

Consider two point charges q1 and q2 separated by a distance d.
The electric field due to the charges at a point P on the line joining them is given by:
- E_total = (kq1) / (r_1^2) + (kq2) / (r_2^2)
Here, r_1 is the distance between q1 and P, r_2 is the distance between q2 and P, k is the Coulomb constant, and q1 and q2 are the magnitudes of the charges.

Electric Field due to a Solid Cone of Charge

Consider a solid cone of charge with height h, radius R, and total charge Q.
The electric field due to the solid cone at a point P on the axis of the cone is given by:
- E = (kQ / hε₀)(1 - (r / √(r^2 + h^2)))
Here, Q is the total charge on the cone, h is the height of the cone, r is the distance from the apex of the cone to point P, and ε₀ is the permittivity of free space.

Electric Field due to an Infinite Line of Charge

Consider an infinite line of charge with linear charge density λ.
The electric field due to the infinite line of charge at a point P at a distance r from the line is given by:
- E = (λ / 2πε₀r)
Here, λ is the charge per unit length along the line charge, r is the distance from the line charge to the point P, and ε₀ is the permittivity of free space.

Electric Field due to an Equilateral Triangle of Charge

Consider an equilateral triangle of charge with side length a and total charge Q.
The electric field due to the equilateral triangle at a point P on the perpendicular bisector of the triangle is given by:
- E = (kQ / (2πε₀a))(1 / (r^2 + (a/√3)^2))
Here, Q is the total charge on the triangle, a is the side length of the triangle, r is the distance from the center of the triangle to point P, k is the Coulomb constant, and ε₀ is the permittivity of free space.