Concepts to remember on Field Due To Dipole And Continuous Charge Distributions:

Electric field due to a dipole:

Definition of electric dipole:

• An electric dipole is a pair of equal and opposite charges separated by a small distance.

Expression for electric field due to an electric dipole at a point on the axial line: $$E_{axial}=\frac{1}{4\pi {ϵ}_0}\frac{2qs}{r^3}$$ where q is the magnitude of each charge, s is the separation distance between the charges, r is the distance from the center of the dipole to the point on the axial line, and (\epsilon_0) is the permittivity of free space.

Expression for electric field due to an electric dipole at a point on the equatorial line: $$E_{equatorial}=\frac{1}{4\pi {ϵ}_0}\frac{qs}{r^3}$$ where q is the magnitude of each charge, s is the separation distance between the charges, r is the distance from the center of the dipole to the point on the equatorial line, and (\epsilon_0) is the permittivity of free space.

Electric potential due to an electric dipole: $$\varphi =\frac{1}{4\pi {ϵ}_0}\frac{2qs}{r}$$ where q is the magnitude of each charge, s is the separation distance between the charges, r is the distance from the center of the dipole to the point where the potential is being calculated, and (\epsilon_0) is the permittivity of free space.

Electric field due to a continuous charge distribution: Concept of continuous charge distribution:

• A continuous charge distribution is a distribution of charge in which the charge is spread out over a continuous region of space, rather than being concentrated at discrete points.

Linear charge distribution: Expression for electric field due to a uniformly charged thin rod: $$E=\frac{1}{4\pi {ϵ}_0}\frac{2\lambda }{r}$$ where (\lambda) is the linear charge density (charge per unit length), r is the distance from the rod to the point where the electric field is being calculated, and (\epsilon_0) is the permittivity of free space.

Surface charge distribution: Expression for electric field due to a uniformly charged thin spherical shell: $$E=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}$$ where Q is the total charge on the shell, r is the distance from the center of the shell to the point where the field is being calculated, and (\epsilon_0) is the permittivity of free space.

Volume charge distribution: Expression for electric field due to a uniformly charged sphere: $$E=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}\left(\frac{3R^2-r^2}{2R^3}\right)$$ for r < R $$E=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}$$ for r > R where Q is the total charge on the sphere, R is the radius of the sphere, r is the distance from the center of the sphere to the point where the field is being calculated, and (\epsilon_0) is the permittivity of free space.

Gauss’s law: Statement of Gauss’s law:

• Gauss’s law states that the net electric flux through any closed surface is equal to the total charge enclosed by the surface.

Mathematical form of Gauss’s law: $$\oint \overrightarrow{E}\cdot \hat{n}dA=\frac{Q_{enc}}{{ϵ}_0}$$ where (\overrightarrow{E}) is the electric field vector, (\hat{n}) is a unit vector perpendicular to the surface, dA is an element of surface area, Q_(enc) is the total charge enclosed by the surface, and (\epsilon_0) is the permittivity of free space.

Applications of Gauss’s law: Finding the electric field due to symmetric charge distributions.

Equipotential surfaces:

Definition of an equipotential surface:

• An equipotential surface is a surface on which the electric potential is the same at every point.

Properties of equipotential surfaces:

• Equipotential surfaces are always perpendicular to the electric field lines.
• No work is done in moving a charge along an equipotential surface.

Relation between electric field and equipotential surfaces:

• The electric field at a point is always perpendicular to the equipotential surface passing through that point.

Electric flux: Definition of electric flux:

• Electric flux is a measure of the amount of electric field passing through a given surface.

Mathematical expression for electric flux: $${\mathrm{\Phi }}_E=\oint \overrightarrow{E}\cdot \hat{n}dA$$ where (\overrightarrow{E}) is the electric field vector, (\hat{n}) is a unit vector perpendicular to the surface, and dA is an element of surface area.

Relation between electric flux and Gauss’s law:

• Gauss’s law states that the net electric flux through any closed surface is equal to the total charge enclosed by the surface.