1. Electric Potential & Potential Difference

Electric Potential :: The electric potential at any point in an electric field is defined as the work done in bringing a unit positive test charge from infinity to that point without acceleration.

Formula: V = W / q0

SI Unit: Volt (V) or Joule/Coulomb

Dimensional Formula: [M L^2 T^-3 A^-1]

Potential Difference :: The potential difference between two points in an electric field is defined as the work done in bringing a unit positive charge from one point to another.

Formula: V_B - V_A = W_AB / q0

Electric Potential due to a Point Charge :: V = (1 / 4piepsilon_0) * (q / r)

Electric Potential due to a Short Electric Dipole ::

At its axis: V = (1 / 4piepsilon_0) * (p / r^2)

At its equatorial position: V = 0

At a general arbitrary point: V = (1 / 4piepsilon_0) * (p cos(theta) / r^2)

Electric Potential due to a System of Charges :: V = V1 + V2 + ... + VN = (1 / 4piepsilon_0) * [(q1/r1) + (q2/r2) + ... + (qN/rN)]

2. Relation Between Electric Field & Potential

Potential Gradient :: The relation expressing the electric field as the negative space rate of change of electric potential.

Formula: E = -dV / dr

3. Electric Potential Energy (E.P.E.)

Electric Potential Energy :: The energy stored within a system of electric charges due to their positions relative to each other within an electric field, equivalent to the work done in slowly bringing the charges from infinity to their respective positions.

E.P.E. of a System of Two Point Charges :: U = (1 / 4piepsilon_0) * (q1 * q2 / r12)

Work Done in Rotating a Dipole :: The work required to change the alignment of an electric dipole in a uniform electric field from an initial angle to a final angle.

Formula: W = pE * (cos(theta_1) - cos(theta_2))

Potential Energy of a Dipole :: U = -pE cos(theta)

Stable Equilibrium: U_min = -pE

Unstable Equilibrium: U_max = pE

4. Electrostatics of Conductors & Spheres

Electrostatic Properties of a Charged Conductor ::

Inside a conductor, the electrostatic field is zero (E = 0).

The interior of a conductor has no excess charge.

Electrostatic potential is constant throughout the volume of the conductor (V = constant).

Electric field at the surface of a charged conductor: E = sigma / epsilon_0.

Potential of a Charged Conducting Sphere / Shell ::

Outside the sphere: V = kq / r

At the surface: V = kq / R

Inside the sphere: V = kq / R

Potential of a Charged Ring ::

At the center: V = kQ / R

At an axial point distance x: V = kQ / square_root(R^2 + x^2)

5. Equipotential Surfaces

Equipotential Surface :: A surface having the same electric potential at each point across it.

Core Properties of Equipotential Surfaces ::

No work is done in moving a test charge over an equipotential surface.

Electric field lines meet an equipotential surface at right angles.

Equipotential surfaces are closer together in regions of strong field and farther apart in weak fields.

No two equipotential surfaces can ever intersect each other.

6. Dielectrics & Polarization

Dielectrics :: Insulating materials containing atoms or molecules whose charge distribution can be altered by an external field.

Non-Polar Dielectrics :: Dielectrics where the positive and negative charge centers coincide, resulting in zero permanent dipole moment (Example: CO2).

Polar Dielectrics :: Dielectrics where the centers of positive and negative charges do not coincide, providing a permanent electric dipole moment (Example: H2O).

Electric Polarization :: The process by which the positive and negative charge centers within a dielectric split or separate further when placed under an external electric field, creating an induced dipole moment.

7. Capacitors & Capacitance

Capacitance :: The capability of a system of conductors to store or hold electric charge.

Formula: C = q / V

SI Unit: Farad (F) or Coulomb/Volt

Capacitance of an Isolated Spherical Conductor :: C = 4piepsilon_0*R

Parallel Plate Capacitor :: A system storing charge consisting of two parallel metallic plates of area A separated by a distance d.

Formula (Vacuum): C = (epsilon_0 * A) / d

Formula (With Dielectric K): C = (K * epsilon_0 * A) / d

Formula (Partially Filled with Dielectric of thickness t): C = (epsilon_0 * A) / [d - t + (t / K)]

8. Combinations of Capacitors

Series Combination :: A connection setup where the charge on each capacitor remains identical while the total potential difference splits across them.

Formula: 1/C_total = 1/C1 + 1/C2 + 1/C3

Conditions: Charge is the same; Total Voltage = V1 + V2 + V3

Parallel Combination :: A connection setup where the potential difference across each capacitor remains identical while the total charge splits across them.

Formula: C_total = C1 + C2 + C3

Conditions: Voltage is the same; Total Charge = q1 + q2 + q3

9. Energy Stored in a Capacitor & Energy Density

Energy Stored in a Capacitor :: U = (1/2) * C * V^2 = (1/2) * q * V = q^2 / (2C)

Energy Density :: The electrostatic potential energy stored per unit volume in the medium between the plates of the capacitor.

Formula: u = (1/2) * epsilon_0 * E^2

Work Done by a Battery :: W = C * V^2 = Q * V

Energy Loss During Sharing :: Loss = (1/2) * C * V^2 (dissipated entirely as heat)

10. Redistribution of Charges

Common Potential :: The unified potential achieved when two isolated conductors of different potentials are connected together via a conducting wire.

Formula: V = (C1V1 + C2V2) / (C1 + C2)

Post-sharing charges: q1 = C1V and q2 = C2V