Concepts

• Electric Field due to an Infinitely Long Straight Wire

• Visualize the wire as a source of electric fields radiating outward in all directions.
• The field lines are parallel to the wire and their density decreases with the square of the distance from the wire.
• Use the formula $$E = \frac{2 k_e \lambda}{r}$$, where $$k_e$$ is the Coulomb constant, (\lambda) is the linear charge density, and $$r$$ is the distance from the wire.

• Electric Field due to a Thin Spherical Shell

• Imagine concentric field lines emanating radially from the center of the spherical shell.
• The field outside the shell is the same as that of a point charge of the same magnitude located at the center.
• Inside the shell, the electric field is zero.
• Use the formula $$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$, where $$\epsilon_0$$ is the vacuum permittivity, $$Q$$ is the total charge enclosed by the shell, and $$r$$ is the distance from the center of the shell.

• Electric Field due to a Uniformly Charged Solid Sphere

• The field has a non-zero value both inside and outside the sphere.
• The field outside the sphere is the same as that of a point charge of the same magnitude located at the center of the sphere.
• The field inside the sphere increases linearly with the distance from the center.
• Use the formulas $$E_{outside} = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$ for points outside the sphere and $$E_{inside} = \frac{1}{4\pi\epsilon_0}\frac{Qr}{R^3}$$ for points inside the sphere, where $$R$$ is the radius of the sphere.

• Gauss’s Law and Its Mathematical Form

• Gauss’s law states that the net flux of the electric field through any closed surface is proportional to the net charge enclosed by that surface.
• Mathematically, $$\oint\overrightarrow{E}\cdot\hat{n}\text{d}A=\frac{Q_{enc}}{\epsilon_0}$$, where $$\hat{n}$$ is the unit normal vector perpendicular to the surface, $$\text{d}A$$ is the differential area element, and $$Q_{enc}$$ is the net charge enclosed by the surface.

• Applications of Gauss’s Law

• Calculate the electric field of various charge distributions by choosing appropriate Gaussian surfaces.
• Simplify complex charge distributions by enclosing them with symmetric Gaussian surfaces that can simplify the calculation.
• Determine the net charge enclosed by a surface by calculating the flux of the electric field through that surface.

• Concept of Flux of the Electric Field

• Flux measures the amount of electric field passing through a surface.
• It is defined as the dot product of the electric field and the surface area vector $$(\overrightarrow{E}\cdot\hat{n})\text{d}A$$.
• The net flux of the electric field through a closed surface is equal to the net charge enclosed by that surface divided by the vacuum permittivity.

• Divergence of the Electric Field

• Divergence measures the net outward flow of the electric field from a point in space.
• It is calculated as the limit of the net flux through a small surface divided by the volume of the enclosed region as the volume approaches zero.
• Divergence is a source term and is related to the charge density by the equation $$\overrightarrow{\nabla}\cdot\overrightarrow{E}=\frac{\rho}{\epsilon_0}$$

• Relation between Divergence of the Electric Field and Charge Density

• Divergence of the electric field at a point is directly proportional to the charge density at that point.
• A positive divergence indicates a source of electric field lines (positive charge), while a negative divergence indicates a sink (negative charge).
• Zero divergence implies the absence of any net charge or a balanced distribution of positive and negative charges.