Potential Due To Different Charge Distributions - Relation Between Electric Field And Potential

Slide 1: Electric Fields and Potentials

Electric field (E) and electric potential (V) are two fundamental concepts in the study of electromagnetism.
The electric field represents the force per unit charge experienced by a test charge at any given point in space.
The electric potential represents the electric potential energy per unit charge at a given point in space.
The electric field and potential are related by the equation: E = -∇V, where ∇ denotes the gradient operator.
In this lecture, we will explore the relation between electric fields and potentials due to different charge distributions.

Slide 2: Electric Potential Due to a Point Charge

The electric potential (V) due to a point charge (Q) can be calculated using the equation: V = kQ/r, where k is the Coulomb’s constant and r is the distance from the charge.
The electric potential decreases with increasing distance from the charge.
The electric field due to a point charge can be calculated by taking the negative gradient of the potential: E = -∇V = kQ/r^2.
The electric field points radially outward for a positive charge and radially inward for a negative charge.
Example: Calculate the electric potential and field due to a +2 μC charge at a distance of 0.5 meters.

Slide 3: Electric Potential Due to a Continuous Charge Distribution

To find the electric potential due to a continuous charge distribution, we integrate the contributions from infinitesimally small charge elements.
The equation for electric potential due to a continuous charge distribution is given by: V = ∫(k dQ)/r, where the integral is taken over the entire charge distribution.
The electric potential decreases with increasing distance from the charge distribution.
Example: Calculate the electric potential at a point P due to a uniformly charged rod of length L and total charge Q.
The electric field due to a continuous charge distribution can be obtained by taking the negative gradient of the potential: E = -∇V.

Slide 4: Electric Potential Due to a Uniformly Charged Sphere

For a uniformly charged sphere, the electric potential at a point outside the sphere is the same as that due to a point charge located at the center of the sphere.
The electric potential inside a uniformly charged sphere is constant and equal to the potential at its surface.
Example: Calculate the electric potential at a point outside a uniformly charged sphere of radius R and total charge Q.
For a conducting sphere, the electric potential inside the sphere is constant and equal to the potential at its surface.
The electric field inside a uniformly charged sphere is zero.

Slide 5: Electric Potential Due to Multiple Point Charges

The electric potential due to multiple point charges is the sum of the potentials due to each individual charge.
The equation for the electric potential due to multiple point charges is given by: V = Σ(kQ_i)/r_i, where the summation is taken over all the charges.
The electric field due to multiple point charges can be obtained by taking the negative gradient of the potential: E = -∇V.
The electric field due to multiple point charges is the vector sum of the individual electric fields.
Example: Calculate the electric potential and field at a given point due to two point charges.

Slide 6: Electric Potential Due to a Line Charge

The electric potential due to a uniformly charged line can be calculated using the equation: V = λ/(2πε₀) * ln(r/r₀), where λ is the linear charge density, ε₀ is the permittivity of free space, r is the distance from the line, and r₀ is a reference distance.
The electric potential decreases logarithmically with increasing distance from the line.
The electric field due to a line charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field points radially outward from a positively charged line and radially inward towards a negatively charged line.
Example: Calculate the electric potential and field at a distance r from a uniformly charged line segment.

Slide 7: Electric Potential Due to a Ring of Charge

The electric potential due to a uniformly charged ring can be calculated using the equation: V = kQ/(√(R² + r²) - √(R² + r₀²)), where Q is the total charge, R is the radius of the ring, r is the distance from the center of the ring, and r₀ is a reference distance.
The electric potential decreases with increasing distance from the ring.
The electric field due to a ring charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero at the center of the ring and is directed radially outward or inward depending on the sign of the charge.
Example: Calculate the electric potential and field at a point on the axis of a uniformly charged ring.

Slide 8: Electric Potential Due to a Disc of Charge

The electric potential due to a uniformly charged disc can be calculated using the equation: V = (σ/2ε₀) * (1 - √(1 + r²/R²) + ln(r/σ)), where σ is the surface charge density, ε₀ is the permittivity of free space, r is the distance from the center of the disc, and R is the radius of the disc.
The electric potential decreases with increasing distance from the center of the disc.
The electric field due to a disc charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field points radially outward from the face of the disc and is zero inside the disc.
Example: Calculate the electric potential and field at a point on the axis of a uniformly charged disc.

Slide 9: Electric Potential Due to a Spherical Shell of Charge

The electric potential due to a uniformly charged spherical shell can be calculated using the equation: V = (kQ/R) * (1/r - 1/R), where Q is the total charge, R is the radius of the shell, and r is the distance from the center of the shell.
The electric potential decreases with increasing distance from the shell.
The electric field due to a spherical shell charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero inside the shell, and outside the shell, it behaves as if the entire charge is concentrated at the center.
Example: Calculate the electric potential and field at a point inside and outside a uniformly charged spherical shell.

Slide 10: Electric Potential Due to a Solid Sphere of Charge

The electric potential due to a uniformly charged solid sphere can be calculated using the equation: V = (3/5) * (kQ/R) * (1 - (r²/R²)), where Q is the total charge, R is the radius of the sphere, and r is the distance from the center of the sphere.
The electric potential decreases with increasing distance from the center of the sphere.
The electric field due to a solid sphere charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero inside the sphere, and outside the sphere, it behaves as if the entire charge is concentrated at the center.
Example: Calculate the electric potential and field at a point inside and outside a uniformly charged solid sphere.

Slide 11: Potential Due To Different Charge Distributions - Relation between electric field and potential

The electric field and electric potential are related by the equation: E = -∇V, where E is the electric field, V is the electric potential, and ∇ denotes the gradient operator.
The gradient of a scalar function gives the rate of change of the function with respect to position. In this case, it represents the direction and magnitude of the electric field.
Electric field is a vector quantity, while electric potential is a scalar quantity.
The electric field is a measure of the force experienced by a test charge, while the electric potential is a measure of the potential energy per unit charge at a point in an electric field.
The negative sign in the equation signifies that the electric field points in the direction of decreasing electric potential.

Slide 12: Potential Due To Different Charge Distributions - Electric Field and Potential due to a Point Charge

Electric Potential due to a point charge (Q) is given by the formula: V = kQ/r, where k is Coulomb’s constant and r is the distance between the charge and the point where potential is calculated.
Electric Field due to a point charge (Q) is given by the formula: E = kQ/r^2.
The electric potential decreases with increasing distance from the point charge.
The electric field points radially outward for a positive charge and radially inward for a negative charge.
Example: Calculate the electric potential and field due to a +4 μC charge at a distance of 2 meters.

Slide 13: Potential Due To Different Charge Distributions - Electric Potential due to a Continuous Charge Distribution

Electric Potential due to a continuous charge distribution is obtained by integrating the potential contributions of infinitesimally small charge elements.
The formula for Electric Potential due to a continuous charge distribution is given by: V = ∫ (k dQ)/r, where the integral is taken over the entire charge distribution.
The electric potential decreases with increasing distance from the charge distribution.
Example: Calculate the electric potential at a point P due to a uniformly charged rod of length L and total charge Q.
The electric field due to a continuous charge distribution can be obtained by taking the negative gradient of the potential: E = -∇V.

Slide 14: Potential Due To Different Charge Distributions - Electric Potential due to a Uniformly Charged Sphere

For a uniformly charged sphere, the electric potential at a point outside the sphere is the same as that due to a point charge located at the center of the sphere.
The electric potential inside a uniformly charged sphere is constant and equal to the potential at its surface.
Example: Calculate the electric potential at a point outside a uniformly charged sphere of radius R and total charge Q.
For a conducting sphere, the electric potential inside the sphere is constant and equal to the potential at its surface.
The electric field inside a uniformly charged sphere is zero.

Slide 15: Potential Due To Different Charge Distributions - Electric Potential due to Multiple Point Charges

Electric Potential due to multiple point charges is the algebraic sum of the potentials due to individual charges.
The formula for Electric Potential due to multiple point charges is given by: V = Σ (kQ_i)/r_i, where the summation is taken over all the charges.
The electric potential decreases with increasing distance from the charges.
The electric field due to multiple point charges is obtained by taking the negative gradient of the potential: E = -∇V.
The electric field due to multiple point charges is the vector sum of the individual electric fields.

Slide 16: Potential Due To Different Charge Distributions - Electric Potential due to a Line Charge

Electric Potential due to a uniformly charged line is given by the formula: V = λ/(2πε₀) * ln(r/r₀), where λ is the linear charge density, ε₀ is the permittivity of free space, r is the distance from the line, and r₀ is a reference distance.
The electric potential decreases logarithmically with increasing distance from the line.
The electric field due to a line charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field points radially outward from a positively charged line and radially inward towards a negatively charged line.
Example: Calculate the electric potential and field at a distance r from a uniformly charged line segment.

Slide 17: Potential Due To Different Charge Distributions - Electric Potential due to a Ring of Charge

Electric Potential due to a uniformly charged ring is given by the formula: V = kQ/(√(R² + r²) - √(R² + r₀²)), where Q is the total charge, R is the radius of the ring, r is the distance from the center of the ring, and r₀ is a reference distance.
The electric potential decreases with increasing distance from the ring.
The electric field due to a ring charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero at the center of the ring and is directed radially outward or inward depending on the sign of the charge.
Example: Calculate the electric potential and field at a point on the axis of a uniformly charged ring.

Slide 18: Potential Due To Different Charge Distributions - Electric Potential due to a Disc of Charge

Electric Potential due to a uniformly charged disc is given by the formula: V = (σ/2ε₀) * (1 - √(1 + r²/R²) + ln(r/σ)), where σ is the surface charge density, ε₀ is the permittivity of free space, r is the distance from the center of the disc, and R is the radius of the disc.
The electric potential decreases with increasing distance from the center of the disc.
The electric field due to a disc charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field points radially outward from the face of the disc and is zero inside the disc.
Example: Calculate the electric potential and field at a point on the axis of a uniformly charged disc.

Slide 19: Potential Due To Different Charge Distributions - Electric Potential due to a Spherical Shell of Charge

Electric Potential due to a uniformly charged spherical shell is given by the formula: V = (kQ/R) * (1/r - 1/R), where Q is the total charge, R is the radius of the shell, and r is the distance from the center of the shell.
The electric potential decreases with increasing distance from the shell.
The electric field due to a spherical shell charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero inside the shell, and outside the shell, it behaves as if the entire charge is concentrated at the center.
Example: Calculate the electric potential and field at a point inside and outside a uniformly charged spherical shell.

Slide 20: Potential Due To Different Charge Distributions - Electric Potential due to a Solid Sphere of Charge

Electric Potential due to a uniformly charged solid sphere is given by the formula: V = (3/5) * (kQ/R) * (1 - (r²/R²)), where Q is the total charge, R is the radius of the sphere, and r is the distance from the center of the sphere.
The electric potential decreases with increasing distance from the center of the sphere.
The electric field due to a solid sphere charge can be calculated by taking the negative gradient of the potential: E = -∇V.
The electric field is zero inside the sphere, and outside the sphere, it behaves as if the entire charge is concentrated at the center.
Example: Calculate the electric potential and field at a point inside and outside a uniformly charged solid sphere.

Slide 21: Potential Due To Different Charge Distributions - Relation between electric field and potential

The electric field and electric potential are related by the equation: E = -∇V, where E is the electric field, V is the electric potential, and ∇ denotes the gradient operator.
The gradient of a scalar function gives the rate of change of the function with respect to position. In this case, it represents the direction and magnitude of the electric field.
Electric field is a vector quantity, while electric potential is a scalar quantity.
The electric field is a measure of the force experienced by a test charge, while the electric potential is a measure of the potential energy per unit charge at a point in an electric field.
The negative sign in the equation signifies that the electric field points in the direction of decreasing electric potential.

Slide 22: Potential Due To Different Charge Distributions - Electric Field and Potential due to a Point Charge

Electric Potential due to a point charge (Q) is given by the formula: V = kQ/r, where k is Coulomb’s constant and r is the distance between the charge and the point where potential is calculated.
Electric Field due to a point charge (Q) is given by the formula: E = kQ/r^2.
The electric potential decreases with increasing distance from the point charge.
The electric field points radially outward for a positive charge and radially inward for a negative charge.
Example: Calculate the electric potential and field due to a +4 μC charge at a distance of 2 meters.

Slide 23: Potential Due To Different Charge Distributions - Electric Potential due to a Continuous Charge Distribution

Electric Potential due to a continuous charge distribution is obtained by integrating the potential contributions of infinitesimally small charge elements.
The formula for Electric Potential due to a continuous charge distribution is given by: V = ∫ (k dQ)/r, where the integral is taken over the entire charge distribution.
The electric potential decreases with increasing distance from the charge distribution.
Example: Calculate the electric potential at a point P due to a uniformly charged rod of length L and total charge Q.
The electric field due to a continuous charge distribution can be obtained by taking the negative gradient of the potential: E = -∇V.

Slide 24: Potential Due To Different Charge Distributions - Electric Potential due to a Uniformly Charged Sphere

For a uniformly charged sphere, the electric potential at a point outside the sphere is the same as that due to a point charge located at the center of the sphere.
The electric potential inside a uniformly charged sphere is constant and equal to the potential at its surface.
Example: Calculate the electric potential at a point outside a uniformly charged sphere of radius R and total charge Q.
For a conducting sphere, the electric potential inside the sphere is constant and equal to the potential at its surface.
The electric field inside a uniformly charged sphere is zero.

Slide 25: Potential Due To Different Charge Distributions - Electric Potential due to Multiple Point Charges

Electric Potential due to multiple point charges is the algebraic sum of the potentials due to individual charges.
The formula for Electric Potential due to multiple point charges is given by: V = Σ (kQ_i)/r_i, where the summation is taken over all the charges.
The electric potential decreases with increasing distance from the charges.
The electric field due to multiple point charges is obtained