Potential Due To Different Charge Distributions - Calculation Of Electric Field From A Given Electric Potentialn

Slide 1: Electric Potential

Electric potential is a scalar quantity denoted by V
It determines the amount of work done when a unit positive charge is moved from a reference point to a specific point in an electric field
The unit of electric potential is Volt (V)
Electric potential is calculated as the ratio of work done to the charge moved: V = W/q

Slide 2: Electric Potential Difference

Electric potential difference is the change in electric potential between two points in an electric field
It is denoted by ΔV
Electric potential difference is also known as voltage
Voltage is measured in volts (V)
It is calculated as the difference in electric potential between the two points: ΔV = V2 - V1

Slide 3: Calculation of Electric Potential Difference

To calculate electric potential difference:

Determine the electric potential at the starting point (V1)

Determine the electric potential at the ending point (V2)

Subtract the electric potential at the starting point from the electric potential at the ending point: ΔV = V2 - V1 Example: If V1 = 10 V and V2 = 20 V, then ΔV = 20 V - 10 V = 10 V

Slide 4: Electric Potential and Energy

Electric potential is related to electric potential energy
Electric potential energy is the energy possessed by a charged particle due to its position in an electric field
The relationship between electric potential energy (PE) and electric potential (V) is given by the equation: PE = qV Example: A charge of 5 C is at a point where the electric potential is 10 V. Its electric potential energy is PE = 5 C * 10 V = 50 J

Slide 5: Equipotential Surfaces

Equipotential surfaces are imaginary surfaces in which the electric potential at every point is the same
Equipotential surfaces are always perpendicular to the electric field lines
The electric field lines and equipotential surfaces are always at right angles to each other
Equipotential surfaces can be visualized as a series of concentric spheres surrounding a point charge

Slide 6: Electric Field and Electric Potential

Electric field and electric potential are related to each other
Electric field (E) is a vector quantity that measures the force experienced by a unit positive charge
Electric potential (V) is a scalar quantity that determines the amount of work done on a unit positive charge
The relationship between electric field and electric potential is given by the equation: E = -∇V

Slide 7: Relationship Between Electric Field and Electric Potential

The negative sign in the equation E = -∇V indicates that the electric field points in the direction of decreasing electric potential
Electric field lines always point from higher potential to lower potential
Electric potential decreases as we move in the direction of the electric field lines Example: If the electric potential decreases from 20 V to 10 V in a distance of 1 meter, then the electric field is given by: E = -(10 V - 20 V)/1 m = -10 V/m

Slide 8: Electric Potential Due to a Point Charge

The electric potential due to a point charge q is given by the equation: V = k(q/r)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
q is the charge of the point charge
r is the distance between the point charge and the point where electric potential is calculated Example: If a point charge of 2 μC is at a distance of 3 meters, then the electric potential due to the point charge is: V = (9 × 10^9 Nm^2/C^2)(2 μC)/(3 m) = (18 × 10^-3 Nm^2/C)/3 m = 6 × 10^6 V

Slide 9: Electric Potential Due to an Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance
The electric potential due to an electric dipole at any point on its axial line is given by the equation: V = k(p/r^2)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
p is the magnitude of the dipole moment (p = q × 2d)
r is the distance from the dipole along its axial line Example: A dipole with a dipole moment of 5 × 10^(-9) Cm is at a distance of 0.1 meters along its axial line. The electric potential due to the dipole is: V = (9 × 10^9 Nm^2/C^2)(5 × 10^(-9) Cm)/(0.1 m^2) = 4.5 × 10^6 V

Slide 10: Electric Potential Due to an Electric Dipole (cont’d)

The electric potential due to an electric dipole at any point on its equatorial line is given by the equation: V = k(p/r^3)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
p is the magnitude of the dipole moment (p = q × 2d)
r is the distance from the dipole along its equatorial line Example: A dipole with a dipole moment of 4 × 10^(-6) Cm is at a distance of 2 meters along its equatorial line. The electric potential due to the dipole is: V = (9 × 10^9 Nm^2/C^2)(4 × 10^(-6) Cm)/(2 m^3) = 9 × 10^(-7) V

Slide 11: Potential Due To Different Charge Distributions

Electric potential can be calculated for various charge distributions
The most common charge distributions include point charges, line charges, surface charges, and volume charges
The formulas for calculating electric potential differ based on the charge distribution
The principles of superposition can be used to determine the total electric potential due to multiple charge distributions

Slide 12: Calculation of Electric Field from a Given Electric Potential

To calculate the electric field from a given electric potential, we use the relationship: E = -∇V
Here, ∇ (del) represents the gradient operator, which is a vector operator that calculates the rate of change of a scalar field
The negative sign indicates that the electric field points in the direction of decreasing potential Example: If the electric potential is given as V = 2x^2 + 3y^2 + 4z^2, then the electric field components can be calculated as: E_x = -∂V/∂x, E_y = -∂V/∂y, E_z = -∂V/∂z

Slide 13: Electric Potential Due to a Continuous Charge Distribution

For continuous charge distributions, such as a line of charge, a surface charge, or a volume charge, the electric potential is obtained by integrating the contribution of infinitesimally small charge elements
The formula for calculating electric potential due to a continuous charge distribution varies based on the specific distribution and geometry Example: To calculate the electric potential due to a ring of charge, we integrate the potential contribution from each infinitesimally small charge element on the ring

Slide 14: Electric Potential Due to a Line of Charge

The electric potential due to a line of charge is given by the equation: V = k(λ/r)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
λ is the linear charge density (charge per unit length) of the line of charge
r is the distance from the line of charge Example: If a line of charge has a linear charge density of 2 μC/m and the distance from the line of charge is 0.5 meters, then the electric potential due to the line of charge is: V = (9 × 10^9 Nm^2/C^2)(2 μC/m)/(0.5 m) = 36 × 10^6 V

Slide 15: Electric Potential Due to a Surface Charge

The electric potential due to a surface charge is given by the equation: V = k(σ/r)
k is the electrostatic constant (9 × 10^9 Nm2/C2)
σ is the surface charge density (charge per unit area) of the surface charge
r is the distance from the surface charge Example: If a surface charge has a surface charge density of 5 μC/m2 and the distance from the surface charge is 1 meter, then the electric potential due to the surface charge is: V = (9 × 10^9 Nm^2/C^2)(5 μC/m^2)/(1 m) = 45 × 10^6 V

Slide 16: Electric Potential Due to a Volume Charge

The electric potential due to a volume charge is given by the equation: V = k(ρ/r)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
ρ is the volume charge density (charge per unit volume) of the volume charge
r is the distance from the volume charge Example: If a volume charge has a volume charge density of 3 μC/m3 and the distance from the volume charge is 0.7 meters, then the electric potential due to the volume charge is: V = (9 × 10^9 Nm^2/C^2)(3 μC/m^3)/(0.7 m) = 38.57 × 10^6 V

Slide 17: Electric Potential Due to a Uniformly Charged Sphere

The electric potential due to a uniformly charged sphere is given by the equation: V = k(q/R) * [1 - (r/R)]
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
q is the total charge of the sphere
R is the radius of the sphere
r is the distance from the center of the sphere Example: If a uniformly charged sphere has a total charge of 6 μC and a radius of 0.5 meters, and the distance from the center of the sphere is 0.3 meters, then the electric potential due to the sphere is: V = (9 × 10^9 Nm^2/C^2)(6 μC)/(0.5 m) * [1 - (0.3 m/0.5 m)] = 28.8 × 10^6 V

Slide 18: Electric Potential Due to a Dipole Along Its Axis

The electric potential due to an electric dipole along its axis is given by the equation: V = k(p/r^2)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
p is the magnitude of the dipole moment (p = 2qd)
r is the distance from the dipole along its axis Example: If an electric dipole has a dipole moment of 8 × 10^(-6) Cm and the distance from the dipole along its axis is 0.1 meter, then the electric potential due to the dipole is: V = (9 × 10^9 Nm^2/C^2) * (8 × 10^(-6) Cm) / (0.1 m^2) = 7.2 × 10^6 V

Slide 19: Electric Potential Due to a Dipole Along Its Equatorial Line

The electric potential due to an electric dipole along its equatorial line is given by the equation: V = k(p/r^3)
k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
p is the magnitude of the dipole moment (p = 2qd)
r is the distance from the dipole along its equatorial line Example: If an electric dipole has a dipole moment of 4 × 10^(-5) Cm and the distance from the dipole along its equatorial line is 0.2 meter, then the electric potential due to the dipole is: V = (9 × 10^9 Nm^2/C^2) * (4 × 10^(-5) Cm) / (0.2 m^3) = 4.5 × 10^5 V

Slide 20: Summary

Electric potential quantifies the work done to move a unit positive charge
Electric potential difference is the change in electric potential between two points
Electric potential differs based on charge distributions (point charges, line charges, surface charges, volume charges)
Calculation of electric potential involves integrating the contributions of infinitesimally small charge elements
Electric potential can be used to determine the electric field using the relationship E = -∇V

Slide 21: Electric Potential Due to Different Charge Distributions

Point charges: The electric potential due to a point charge is given by the equation V = k(q/r), where q is the charge of the point charge and r is the distance from the point charge.
Line charges: The electric potential due to a line charge is given by the equation V = k(λ/r), where λ is the linear charge density (charge per unit length) of the line charge and r is the distance from the line charge.
Surface charges: The electric potential due to a surface charge is given by the equation V = k(σ/r), where σ is the surface charge density (charge per unit area) of the surface charge and r is the distance from the surface charge.
Volume charges: The electric potential due to a volume charge is given by the equation V = k(ρ/r), where ρ is the volume charge density (charge per unit volume) of the volume charge and r is the distance from the volume charge. Example: For a sphere with a radius of 0.1 m and a total charge of 4 μC distributed uniformly, the volume charge density is ρ = (4 μC) / [(4/3)π(0.1 m)^3]. If the distance from the center of the sphere is 0.05 m, the electric potential can be calculated using the formula V = k(ρ/r).

Slide 22: Calculation of Electric Field from a Given Electric Potential

To calculate the electric field from a given electric potential, we can use the equation E = -∇V, where ∇ is the gradient operator.
The gradient of a scalar field is a vector that points in the direction of the maximum rate of change of the scalar field.
In three dimensions, the gradient operator is given by ∇ = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k, where i, j, and k are unit vectors in the x, y, and z directions respectively. Example: If the electric potential is given as V(x, y, z) = 2x^2 + 3y^2 + 4z^2, then the electric field components can be calculated as E_x = -∂V/∂x, E_y = -∂V/∂y, E_z = -∂V/∂z.

Slide 23: Calculation of Electric Field from a Given Electric Potential (cont’d)

The electric field can also be calculated using the relationship E = -∇V.
In one dimension, the gradient reduces to the ordinary derivative with respect to the position coordinate. Example: If the electric potential is given as V(x) = 3x^2 + 4x, then the electric field E(x) can be calculated as E(x) = -dV/dx = -6x - 4.

Slide 24: Electric Potential and Electric Field for Point Charges

For a point charge q, the electric potential is given by V = k(q/r), where r is the distance from the point charge.
The electric field due to a point charge q is given by E = k(q/r^2), where r is the distance from the point charge. Example: For a point charge of 2 μC located at the origin, the electric potential and electric field at a distance of 0.5 m can be calculated using the formulas V = k(q/r) and E = k(q/r^2).

Slide 25: Electric Potential and Electric Field for Line Charges

For a line charge with linear charge density λ, the electric potential is given by V = k(λ/r), where r is the distance from the line charge.
The electric field due to a line charge with linear charge density λ is given by E = kλ/(2πr), where r is the distance from the line charge. Example: For a line charge with a linear charge density of 3 μC/m located along the x-axis, the electric potential and electric field at a distance of 2 m along the y-axis can be calculated using the formulas V = k(λ/r) and E = kλ/(2πr).

Slide 26: Electric Potential and Electric Field for Surface Charges

For a surface charge with surface charge density σ, the electric potential is given by V = k(σ/r), where r is the distance from the surface charge.
The electric field due to a surface charge with surface charge density σ is given by E = kσ/(2ε₀), where ε₀ is the permittivity of free space. Example: For a surface charge with a surface charge density of 5 μC/m^2 located on the xy-plane, the electric potential and electric field at a distance of 1 m along the z-axis can be calculated using the formulas V = k(σ/r) and E = kσ/(2ε₀).

Slide 27: Electric Potential and Electric Field for Volume Charges

For a volume charge with volume charge density ρ, the electric potential is given by V = k(ρ/r), where r is the distance from the volume charge.
The electric field due to a volume charge with volume charge density ρ is given by E = kρr/(3ε₀), where ε₀ is the permittivity of free space. Example: For a volume charge with a volume charge density of 2 μC/m^3 located