### Field Due To Dipole And Continuous Charge Distributions

**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:**
$$E_{axial} = \frac{1}{4\pi\epsilon_0}\frac{2qs}{r^3}$$
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:**
$$E_{equatorial} = \frac{1}{4\pi\epsilon_0}\frac{qs}{r^3}$$
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:**
$$\phi = \frac{1}{4\pi\epsilon_0}\frac{2qs}{r}$$
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:**
$$E = \frac{1}{4\pi\epsilon_0}\frac{2\lambda}{r}$$
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:**
$$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$
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**:
$$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\left(\frac{3R^2-r^2}{2R^3}\right)$$
for r < R
$$E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$$
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:**
$$\oint\overrightarrow{E}\cdot\hat{n}dA=\frac{Q_{enc}}{\epsilon_0}$$
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:**
$$\Phi_E=\oint\overrightarrow{E}\cdot\hat{n}dA$$
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.