Conductors and Electric Potential

• Electric potential inside a conductor:
  • In electrostatic equilibrium, the electric potential inside a conductor is constant.
• Electric potential of a hollow conducting sphere:
  • The electric potential inside a hollow conducting sphere is constant and equal to the potential of the surface.
• Electric potential of a solid conducting sphere:
  • The electric potential inside a solid conducting sphere increases as we move closer to the center.
• Electric potential due to a charged spherical shell:
  • Inside the shell, potential is constant and equal to the potential of the shell.
  • Outside the shell, potential decreases as we move away from the shell.
• Electric potential of a capacitor:
  • The electric potential difference between the plates of a capacitor is equal to the potential difference applied across it.
• Equipotential surfaces in a capacitor:
  • The plates of a capacitor are equipotential surfaces.
  • Equipotential surfaces are parallel to the plates.

Electric Dipole

• Definition:
  • An electric dipole consists of two equal and opposite charges separated by a small distance.
• Electric field due to an electric dipole:
  • The electric field lines point from positive to negative charge.
  • Along the axis of the dipole, the electric field decreases with increasing distance.
  • Along the perpendicular bisector of the dipole, the electric field is zero.
• Potential due to an electric dipole:
  • The potential at a point due to an electric dipole is given by V = (kp cosθ)/r², where kp is the product of the electric dipole moment and charge, θ is the angle between the dipole axis and the line joining the point with the center of the dipole, and r is the distance from the center of the dipole.
• Torque on an electric dipole:
  • The torque experienced by an electric dipole in an electric field is given by τ = pE sinθ, where p is the electric dipole moment, E is the electric field strength, and θ is the angle between the dipole moment and the electric field.
• Potential energy of an electric dipole in an electric field:
  • The potential energy of an electric dipole in an electric field is given by U = -pE cosθ, where U is the potential energy, p is the electric dipole moment, E is the electric field strength, and θ is the angle between the dipole moment and the electric field.
• Electric field due to a dipole in axial and equatorial positions:
  • In the axial position, the electric field is stronger and points along the axis of the dipole.
  • In the equatorial position, the electric field is weaker and points away from the dipole axis.

Gauss’s Law and its Applications

• Gauss’s law:
  • The electric flux through any closed surface is equal to the total charge enclosed by that surface divided by the permittivity of the medium.
• Electric field due to a uniformly charged spherical shell:
  • Inside the shell, the electric field is zero.
  • Outside the shell, the electric field is the same as that due to a point charge at the center.
• Electric field due to a non-uniformly charged spherical shell:
  • The electric field at a point outside a charged, non-uniformly charged spherical shell can be found using Gauss’s law.
• Electric field due to a uniformly charged solid sphere:
  • The electric field inside the sphere varies with the distance from the center.
  • The electric field outside the sphere is the same as that due to a point charge at the center.
• Electric field due to an infinite plane sheet of charge:
  • The electric field is uniform and directed away from the sheet on both sides.
• Electric field due to a uniformly charged thin straight wire:
  • The electric field strength decreases with increasing distance from the wire.

11. Electric Potential Energy

• Electric potential energy is the stored energy of a system of charges.
• It is given by the equation: PE = qV, where PE represents potential energy, q is the charge, and V is the electric potential.
• The unit of potential energy is joule (J).
• Positive potential energy corresponds to charges that repel each other, while negative potential energy corresponds to charges that attract each other.
• The potential energy between two point charges can be calculated using the equation: PE = (kq₁q₂) / r, where k is the Coulomb’s constant and r is the distance between the charges.

12. Potential Energy of a System of Charges

• When multiple charges are present in a system, the total potential energy is the algebraic sum of the potential energies of individual charges.
• The potential energy of a system of charges depends on their configuration and relative positions.
• A configuration with maximum potential energy is said to be in an unstable equilibrium, while a configuration with minimum potential energy is in a stable equilibrium.
• Example: Two positive charges of equal magnitude brought together will have potential energy that increases to the maximum when they are at the closest distance.
• Example: A positive and negative charge brought closer will have potential energy that decreases to the minimum when they are at the closest distance.

13. Electric Potential Energy and Work

• Work is done when charges are moved against an electric field.
• The work done (W) in moving a charge (q) between two points with different electric potential (V₁ and V₂) is given by the equation: W = q(V₂ - V₁).
• If the potential difference is positive, work is done by an external agent on the charge.
• If the potential difference is negative, work is done by the charge on the external agent.
• The unit of work and electric potential energy is joule (J).

14. Conservative Nature of Electric Force

• The electric force between charges is a conservative force.
• A conservative force depends only on the initial and final positions of objects, not the path taken.
• This means that the work done by or against an electric force is independent of the path taken.
• The work done in moving a charge in a closed loop is zero if no external forces act on the charge.

15. Capacitors and Electric Potential Energy

• A capacitor is a device used to store electric potential energy.
• It consists of two conductive plates separated by a dielectric material.
• When a potential difference (V) is applied across the plates, charges accumulate on the plates, storing electric potential energy.
• The potential energy stored in a capacitor is given by the equation: PE = (1/2)CV², where C is the capacitance and V is the potential difference across the plates.
• Capacitors have various applications, such as energy storage in electronic devices and filtering in electrical circuits.

16. Example: Calculation of Electric Potential Energy

• Consider a system with two charges, q₁ = +3C and q₂ = -2C, separated by a distance of 4m.
• The electrostatic potential energy of this system can be calculated using the equation: PE = (kq₁q₂) / r.
• Substituting the given values, PE = (9 × 10^9 Nm²/C²) × (+3C) × (-2C) / 4m = -13.5 J.
• The negative sign indicates that the charges attract each other, resulting in potential energy that can be released if they move towards each other.

17. Equipotential Surfaces

• An equipotential surface is a surface in an electric field where the electric potential is the same everywhere.
• The electric field is perpendicular to the equipotential surfaces.
• Equipotential surfaces are represented by dashed lines on diagrams.
• Equipotential surfaces become closer together where the electric field is stronger and are farther apart where the electric field is weaker.
• Electric field lines are always perpendicular to equipotential surfaces.

18. Electric Potential Gradient

• Electric potential gradient is the rate of change of electric potential with distance.
• It is calculated using the equation: E = -dV/dr, where E is the electric field strength, dV is the change in electric potential, and dr is the change in distance.
• The electric field points in the direction of decreasing potential.
• Electric field lines are always perpendicular to equipotential surfaces because equipotential surfaces have constant potential, indicating no change in potential along the surface.
• The magnitude of the electric field is greater where the equipotential lines are closer together.

19. Electric Potential and Conductors

• In electrostatic equilibrium, the electric potential inside a conductor is constant.
• Conductors have free charges that redistribute themselves until the electric potential is the same everywhere within the conductor.
• The electric field inside a conductor is zero because charges distribute themselves in a manner that cancels out the electric field inside.
• Electric field lines can only exist outside a conductor.
• In case of excess charge on a conductor, the charge distributes itself on the outer surface to minimize electrostatic repulsion.

20. Electric Potential of a Spherical Conductor

• The electric potential inside a hollow conducting sphere is constant and equal to the potential of the surface.
• The electric potential inside a solid conducting sphere increases as we move closer to the center.
• The electric field within a conductor is always perpendicular to its surface.
• Excess charges on a conductor reside on its outer surface.
• Conductors maintain constant potential because excess charge distributes evenly on the surface to create an equipotential surface.

21. Electric Potential due to a Charged Rod

• The electric potential due to a charged rod can be calculated using the equation: V = (kλ / r) ln(b/a), where V is the electric potential, k is the Coulomb’s constant, λ is the charge per unit length on the rod, r is the distance from the rod, a is the starting point of integration, and b is the ending point of integration.
• Example: A uniformly charged rod of length L has charge +Q. Calculate the electric potential at a point P located at a distance r from the center of the rod.
  • Divide the rod into small elemental sections of length dx.
  • The charge on each elemental section is dq = (Q/L) dx.
  • Use the equation V = (kλ / r) ln(b/a) to calculate the potential at point P due to each elemental section.
  • Integrate the potentials over the entire length of the rod to find the total potential at point P.

22. Electric Potential due to a Charged Disk

• The electric potential due to a charged disk can be calculated using the equation: V = (kσ / 2ε₀) (1 - (z / sqrt(z² + R²))), where V is the electric potential, k is the Coulomb’s constant, σ is the charge density on the disk, ε₀ is the permittivity of free space, z is the distance from the center of the disk, and R is the radius of the disk.
• Example: A uniformly charged disk of radius R has a charge density σ. Calculate the electric potential at a point P located on the axis of the disk at a distance z from the center.
  • Divide the disk into small elemental rings of width dz.
  • The charge on each elemental ring is dq = σ (2πr dz), where r is the radius of the elemental ring.
  • Use the equation V = (kσ / 2ε₀) (1 - (z / sqrt(z² + R²))) to calculate the potential at point P due to each elemental ring.
  • Integrate the potentials over the entire disk to find the total potential at point P.

23. Electric Potential due to a Charged Infinite Line of Charge

• The electric potential due to a charged infinite line of charge can be calculated using the equation: V = (kλ / 2πε₀) ln(r / a), where V is the electric potential, k is the Coulomb’s constant, λ is the charge per unit length on the line, r is the distance from the line, and a is a reference distance.
• Example: A uniformly charged infinite line of charge has a charge density λ. Calculate the electric potential at a point P located at a distance r from the line.
  • Divide the line into small elemental sections of length dl.
  • The charge on each elemental section is dq = λ dl.
  • Use the equation V = (kλ / 2πε₀) ln(r / a) to calculate the potential at point P due to each elemental section.
  • Integrate the potentials over the entire line to find the total potential at point P.

24. Electric Potential due to a Charged Ring

• The electric potential due to a charged ring can be calculated using the equation: V = (kQ / 4πε₀) * (1 / sqrt(R² + r² - 2Rr cosθ)), where V is the electric potential, k is the Coulomb’s constant, Q is the total charge on the ring, ε₀ is the permittivity of free space, R is the radius of the ring, r is the distance from the center of the ring to the point at which potential is calculated, and θ is the angle made by r with the axis of the ring.
• Example: A uniformly charged ring of radius R has a total charge Q. Calculate the electric potential at a point P located on the axis of the ring at a distance r from the center.
  • Divide the ring into small elemental sections of length dθ.
  • The charge on each elemental section is dq = (Q / 2πR) dθ.
  • Use the equation V = (kQ / 4πε₀) * (1 / sqrt(R² + r² - 2Rr cosθ)) to calculate the potential at point P due to each elemental section.
  • Integrate the potentials over the entire ring to find the total potential at point P.

25. Capacitance of a Parallel Plate Capacitor

• The capacitance of a parallel plate capacitor can be calculated using the equation: C = ε₀ (A / d), where C is the capacitance, ε₀ is the permittivity of free space, A is the area of one of the plates, and d is the distance between the plates.
• Capacitance is a measure of a capacitor’s ability to store electric charge.
• Capacitance is measured in farads (F), where 1 farad = 1 coulomb / 1 volt.
• The greater the capacitance, the more charge a capacitor can store for a given potential difference.

26. Capacitance of a Cylindrical Capacitor

• The capacitance of a cylindrical capacitor can be calculated using the equation: C = (2πε₀L) / ln(b/a), where C is the capacitance, ε₀ is the permittivity of free space, L is the length of the capacitor, and a and b are the radii of the inner and outer cylinders, respectively.
• Cylindrical capacitors have two concentric cylinders as the plates.
• The capacitance of a cylindrical capacitor depends on the length of the capacitor and the radii of the cylinders.
• Cylindrical capacitors are commonly used in applications such as high energy storage and particle accelerators.

27. Energy Stored in a Capacitor

• The energy stored in a capacitor can be calculated using the equation: U = (1/2)CV², where U is the energy stored, C is the capacitance, and V is the potential difference across the plates of the capacitor.
• The energy stored in a capacitor is also known as the capacitive energy.
• The energy stored in a capacitor is directly proportional to the square of the potential difference across it.
• Capacitors store energy in the form of electric field and can release it when needed.

28. Dielectric Materials and Capacitance

• Dielectric materials are used to increase the capacitance of a capacitor.
• The presence of a dielectric material between the plates of a capacitor decreases the electric field strength, thus increasing the capacitance.
• Dielectric materials have a high dielectric constant, which is the ratio of the permittivity of the material to the permittivity of free space.
• The capacitance of a capacitor with a dielectric material can be calculated using the equation: C’ = kC, where C’ is the new capacitance, k is the dielectric constant, and C is the initial capacitance without the dielectric.

29. Charging and Discharging of a Capacitor

• When a capacitor is connected to a power source, such as a battery, it charges up with time.
• The charging of a capacitor is an exponential process, described by the equation: Q = Q₀(1 - e^(-t/RC)), where Q is the charge on the capacitor at time t, Q₀ is the final charge, R is the resistance in the circuit, and C is the capacitance.
• The discharging of a capacitor is also an exponential process, described by the equation: Q = Q₀e^(-t/RC).
• The time constant (τ) of a charging or discharging process is given by the equation: τ = RC.

30. RC Circuits and Time Constants

• An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel.
• In a series RC circuit, the voltage across the resistor and the capacitor in the circuit change with time.
• The time constant (τ) of an RC circuit is given by the equation: τ = RC.
• The time constant determines the rate at which the voltage across the capacitor or the current in the circuit changes.
• RC circuits have various