Electrostatics

1. Electric charges:

• Positive charges: Protons in an atom carry positive charges.
• Negative charges: Electrons in an atom carry negative charges.
• Conservation of charge: The total charge in an isolated system remains constant. Charge can neither be created nor destroyed.

2. Conductors and insulators:

• Conductors: Materials that allow electric charges to move freely through them.
• Insulators: Materials that do not allow electric charges to move freely through them.

3. Charging by friction, conduction, induction:

• Charging by friction: When two different materials are rubbed together, electrons are transferred from one material to the other, resulting in the charging of the materials.
• Charging by conduction: When a conductor is connected to a charged object, electrons move from the charged object to the conductor, causing the conductor to become charged.
• Charging by induction: When a charged object is brought near a conductor, electrons in the conductor move away from the charged object, creating a region of positive charge on the conductor closest to the charged object. This process is called electrostatic induction.

Coulomb’s Law:

• Coulomb’s law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.
• Mathematical expression of Coulomb’s law:

$$F = k\frac{q_1q_2}{r^2}$$

Where,

• $F$ is the electrostatic force in Newtons (N).
• $k$ is the electrostatic constant ($k = 8.988 \times 10^9$ Nm²/C²).
• $q_1$ and $q_2$ are the magnitudes of the charges in Coulombs (C).
• $r$ is the distance between the charges in meters (m).

Electric Field:

• The electric field at a point is defined as the electric force experienced by a positive test charge placed at that point divided by the magnitude of the test charge.
• Electric field due to a point charge:

$$E = k\frac{q}{r^2}$$

Where,

• $E$ is the electric field in Newtons per Coulomb (N/C).

• $k$ is the electrostatic constant.

• $q$ is the magnitude of the point charge in Coulombs (C).

• $r$ is the distance between the point charge and the point where the electric field is being calculated in meters (m).

• Electric field due to a continuous charge distribution:

$$E = \int\frac{dq}{4\pi\epsilon_0r^2}$$

Where,

• E = electric field

• dq = incremental charge

• r = distance from the source charge to the observation location

• ϵ0 = permittivity of free space

• Electric field lines:

• Electric field lines are imaginary lines drawn in an electric field to represent the direction and strength of the electric field at each point.

• The direction of an electric field line is the direction in which a positive test charge would experience a force.

• The density of electric field lines in a region indicates the strength of the electric field in that region.

Electric Potential:

• Electric potential at a point is defined as the amount of work done per unit positive charge in bringing a positive test charge from infinity to that point.
• Electric potential due to a point charge:

$$V = k\frac{q}{r}$$

Where,

• $V$ is the electric potential in Volts (V).

• $k$ is the electrostatic constant.

• $q$ is the magnitude of the point charge in Coulombs (C).

• $r$ is the distance between the point charge and the point where the electric potential is being calculated in meters (m).

• Potential difference:

Potential difference between two points is the difference in electric potential between those two points.

$$V = V_B - V_A$$

Where,

• V = potential difference

• Vb = electric potential at point B

• Va = electric potential at point A

• Relationship between electric potential and electric field:

$$E = -\frac{dV}{dx}$$

Where,

• E represents electric field
• V represents the electric potential.
• dr is the change in position.

Gauss’s Law:

• Gauss’s law states that the net electric flux through any closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space.
• Mathematical expression of Gauss’s law:

$$\oint \vec{E} \cdot \hat{n}dA = \frac{Q_{in}}{\epsilon_0}$$

Where,

• $\overrightarrow{E}$ is the electric field vector.
• $\hat{n}$ is the unit normal vector perpendicular to the surface.
• $dA$ is the differential of area.
• $Q_{in}$ is the total charge enclosed by the surface.
• $\epsilon_0$ is the permittivity of free space.

Applications:

• Coulomb’s law is used to calculate the electric force between charged objects.
• Electric field and electric potential are used to calculate the electric forces and torques experienced by charged objects in electric fields.
• Gauss’s law is used to calculate the electric flux through a closed surface and the total charge enclosed by that surface.
• Capacitors are used to store electrical energy.
• Dielectric materials are used to increase the capacitance of capacitors.

Capacitance:

• Capacitance of a parallel-plate capacitor:

$$C = \frac{\epsilon_0 A}{d}$$

Where,

• $C$ is the capacitance in Farads (F).

• $\epsilon_0$ is the permittivity of free space.

• $A$ is the area of each plate in square meters (m²).

• $d$ is the distance between the plates in meters (m).

• Energy stored in a capacitor:

$$U = \frac{1}{2}CV^2$$

Where,

• $U$ is the energy stored in Joules (J).
• $C$ is the capacitance in Farads (F).
• $V$ is the potential difference across the capacitor in Volts (V).

Dielectrics:

• Dielectric materials are non-conducting materials that can be polarized by an electric field.
• When a dielectric material is placed between the plates of a capacitor, it increases the capacitance of the capacitor.
• The polarization of a dielectric material is the process by which the molecules of the material are aligned with the electric field.

References (NCERT books):

• Electrostatics (Chapter 1) - Class 12 NCERT Physics textbook.
• Electric Potential and Capacitance (Chapter 2) - Class 12 NCERT Physics textbook.