Potential Due To Different Charge Distributions

Slide s 1 to 10:

Recap of electrostatic potential
Introduction to potential due to different charge distributions
Importance and applications of the concept

Point Charges

Calculation of potential due to a point charge
Example: Calculating the potential at a point due to a single point charge
Equation: V = k * q / r, where V is the potential, q is the charge, r is the distance from the charge, and k is the Coulomb constant

Line Charges

Calculation of potential due to a line charge
Example: Calculating the potential at a point due to an infinite line charge
Equation: V = 2 * k * λ * ln(r), where V is the potential, λ is the linear charge density, r is the distance from the line charge, and k is the Coulomb constant

Surface Charges

Calculation of potential due to a surface charge
Example: Calculating the potential at a point due to an infinite plane uniformly charged
Equation: V = k * σ / ε₀, where V is the potential, σ is the surface charge density, ε₀ is the permittivity of free space, k is the Coulomb constant

Volume Charges

Calculation of potential due to a volume charge
Example: Calculating the potential at a point due to a uniformly charged sphere
Equation: V = k * Q / R, where V is the potential, Q is the total charge, R is the distance from the center of the sphere, and k is the Coulomb constant

Potential Due to a Dipole

Introduction to dipole
Calculation of potential due to a dipole
Example: Calculating the potential at a point on the equatorial plane of a dipole
Equation: V = k * p * cos(θ) / r², where V is the potential, p is the dipole moment, θ is the angle between the dipole moment and the line connecting the point and the dipole, r is the distance from the dipole, and k is the Coulomb constant

Potential Due to Multiple Charge Distributions

Calculation of potential due to multiple charge distributions
Example: Calculating the total potential at a point due to a combination of point charges and line charges
Using the principle of superposition to calculate the total potential
Equation: V_total = V₁ + V₂ + … + Vₙ, where V_total is the total potential, V₁, V₂, … are the potentials due to individual charge distributions

Equipotential Surfaces

Introduction to equipotential surfaces
Relationship between electric field and equipotential surfaces
Example: Calculating the electric field at a point using equipotential surfaces
Illustration of equipotential surfaces for various charge distributions

Potential Energy

Introduction to potential energy
Calculation of potential energy
Example: Calculating the potential energy of a system of charges
Relationship between potential energy and work done

Questions and Summary

Review of key concepts covered
Answering questions from students
Recap on potential due to different charge distributions
Importance and applications of the concept

Potential Due To Different Charge Distributions - HW problem (on the concept of potential due to dipole)

Consider a dipole consisting of two point charges, +q and -q, separated by a distance d
The dipole moment (p) is given by the product of charge magnitude (q) and separation distance (d): p = q * d
Find the potential at a point on the equatorial plane of the dipole, located at a distance r from the dipole

Solution to the HW Problem

The potential at a point on the equatorial plane can be calculated using the equation: V = k * p * cos(θ) / r²
In this case, since the point is on the equatorial plane, the angle (θ) between the dipole moment and the line connecting the point and the dipole is 90°
Hence, cos(90°) = 0, and the potential simplifies to: V = 0

Electric Potential Energy

Electric potential energy is the amount of work required to bring a charge from infinity to a specific point in an electric field
It is given by the equation: PE = q * V, where PE is the potential energy, q is the charge, and V is the electric potential at that point

Calculating Electric Potential Energy

To calculate the electric potential energy, multiply the charge (q) by the electric potential (V) at that point
Example: If a charge of 2 C is placed in an electric field with a potential of 12 V, the potential energy of the charge is: PE = 2 C * 12 V = 24 J

Relationship between Electric Potential and Electric Field

Electric potential (V) and electric field (E) are related to each other through the equation: E = -dV/dr
In other words, the electric field is the negative gradient of the electric potential

Relationship between Electric Potential and Electric Field - Example

Consider a uniform electric field E with a magnitude of 10 N/C
If the electric potential at a point is constant at 20 V, calculate the rate of change of potential with respect to distance
Using the equation, E = -dV/dr, we can solve for dV/dr:
10 N/C = -dV/dr
Rearranging, dV = -10 N/C * dr
Integrating both sides, ∫dV = -∫10 N/C * dr
V = -10 C * r + C
Since the potential is constant at 20 V, in terms of dr, dV/dr = 0
0 = -10 N/C
Therefore, the rate of change of potential with respect to distance is 0

Electric Potential and Work Done

Work done (W) in moving a charge q from point A to point B is given by: W = q * (V_B - V_A)
The work done is equal to the change in electric potential energy of the charge
If the potential at point B is higher than at point A, work is done on the charge and the potential energy increases
If the potential at point B is lower than at point A, work is done by the charge and the potential energy decreases

Electric Potential and Work Done - Example

If a charge of 4 C is moved from point A to point B in an electric field with a potential difference of 10 V, calculate the work done
Using the equation, W = q * (V_B - V_A), we can substitute the given values:
W = 4 C * (10 V - 0 V) = 40 J
Therefore, the work done in moving the charge is 40 J

Electric Potential and Conductors

In electrostatic equilibrium, the electric potential inside a conductor is constant
The electric field inside a conductor is zero, and the electric potential is the same at all points inside the conductor
The electric potential on the surface of a charged conductor is higher where the charge is concentrated and lower where there is less charge

Electric Potential and Conductors - Example

If a conductor has a net charge of 5 C and the potential at a point on the surface of the conductor is 10 V, calculate the potential inside the conductor
Since the potential is constant inside the conductor, the potential inside is also 10 V
Therefore, the potential inside the conductor is 10 V

Electric Potential and Capacitors

A capacitor consists of two conducting plates separated by an insulating material
When a potential difference is applied to the plates, they become charged
The electric potential on each plate is equal in magnitude and opposite in sign

Electric Potential and Capacitors - Equations

The potential difference (V) across a capacitor is given by: V = Q / C, where Q is the charge stored on the capacitor and C is its capacitance
The energy stored in a capacitor is given by: U = (1/2) * C * V²

Electric Potential and Capacitors - Example

Consider a capacitor with a capacitance of 4 F and a charge of 6 C stored on it
Calculate the potential difference across the capacitor and the energy stored in it
Using the equation, V = Q / C:
V = 6 C / 4 F = 1.5 V
Using the equation, U = (1/2) * C * V²:
U = (1/2) * 4 F * (1.5 V)² = 4.5 J
Therefore, the potential difference across the capacitor is 1.5 V and the energy stored in it is 4.5 J

Electric Potential and Spherical Capacitors

A spherical capacitor consists of two concentric conducting spheres separated by an insulating material
The potential difference across the capacitor depends on the charge stored on each sphere and the radius of the spheres

Electric Potential and Spherical Capacitors - Equations

The potential difference (V) across a spherical capacitor is given by: V = k * (Q₁ / r₁ - Q₂ / r₂), where Q₁ and Q₂ are the charges stored on the spheres, r₁ and r₂ are the radii of the spheres, and k is the Coulomb constant
The capacitance (C) of a spherical capacitor is given by: C = 4πε₀ * (r₁ * r₂) / (r₂ - r₁)

Electric Potential and Spherical Capacitors - Example

Consider a spherical capacitor with an inner sphere of radius 2 cm and an outer sphere of radius 4 cm
The charge on the inner sphere is 2 μC and the charge on the outer sphere is 4 μC
Calculate the potential difference across the capacitor and its capacitance
Using the equation, V = k * (Q₁ / r₁ - Q₂ / r₂):
V = 9 x 10^9 Nm²/C² * (2 x 10^-6 C / 0.02 m - 4 x 10^-6 C / 0.04 m) = -45 V
Using the equation, C = 4πε₀ * (r₁ * r₂) / (r₂ - r₁):
C = 4π * 8.85 x 10^-12 F/m * (0.02 m * 0.04 m) / (0.04 m - 0.02 m) = 2.36 x 10^-11 F
Therefore, the potential difference across the capacitor is -45 V and its capacitance is 2.36 x 10^-11 F

Electric Potential and Parallel Plate Capacitors

A parallel plate capacitor consists of two parallel conducting plates separated by a distance
The potential difference across the capacitor is directly proportional to the electric field and the distance between the plates

Electric Potential and Parallel Plate Capacitors - Equations

The potential difference (V) across a parallel plate capacitor is given by: V = Ed, where E is the electric field between the plates and d is the distance between the plates
The capacitance (C) of a parallel plate capacitor is given by: C = ε₀A / d, where ε₀ is the permittivity of free space and A is the area of each plate

Electric Potential and Parallel Plate Capacitors - Example

Consider a parallel plate capacitor with a plate area of 10 cm² and a plate separation of 1 mm
The electric field between the plates is 1000 N/C
Calculate the potential difference across the capacitor and its capacitance
Using the equation, V = Ed:
V = 1000 N/C * 0.001 m = 1 V
Using the equation, C = ε₀A / d:
C = 8.85 x 10^-12 F/m * (0.1 m² / 0.001 m) = 8.85 x 10^-10 F
Therefore, the potential difference across the capacitor is 1 V and its capacitance is 8.85 x 10^-10 F

Summary and Review

Recap of key concepts covered in the lecture
Importance and applications of potential due to different charge distributions
Potential due to point charges, line charges, surface charges, volume charges, and dipoles
Relationship between electric potential and electric field
Electric potential energy and work done
Electric potential in capacitors and different capacitor configurations