Potential Due To Different Charge Distributions - Calculation Of Potential Difference Between Two Points

Slide 1: Potential Due to Different Charge Distributions - Calculation of Potential Difference between Two Points

In this topic, we will learn about the potential due to different charge distributions.
We will focus on calculating the potential difference between two points.
Understanding the concept of electric potential is crucial for exploring the behavior of electric charges.

Slide 2: Electric Potential

Electric potential is the amount of work done in moving a unit positive charge from infinity to a specific point in an electric field.
It is denoted by V and measured in volts (V).
The electric potential at a point depends on the charge distribution and the distance from the point to the source charges.

Slide 3: Electric Potential Formula

The electric potential at a point due to a point charge q is given by the equation: V = k * (q / r)

In this equation:

V represents the electric potential at the point
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
q is the charge of the point source
r is the distance between the point source and the point where potential is measured

Slide 4: Potential Difference

Potential difference (ΔV) is the difference in electric potential between two points.
It measures the amount of work needed to move a positive unit charge between the points.
It is measured in volts (V) and can be calculated using the formula: ΔV = V2 - V1

In this equation:

ΔV represents the potential difference between the two points
V2 is the electric potential at the second point
V1 is the electric potential at the first point

Slide 5: Calculation of Potential Difference

To calculate the potential difference between two points due to charge distributions, follow these steps:

Identify the charge distribution and the two points.

Calculate the electric potential at each point using the relevant equations.

Subtract the electric potential at the first point from the electric potential at the second point to obtain the potential difference. Example:

Let’s say we have a point charge of +2 μC at position P and another point charge of +4 μC at position Q. We want to find the potential difference between these points.
Given:
- Charge at P: +2 μC
- Charge at Q: +4 μC
Steps:
1. Calculate the electric potential at point P using the relevant equation.
2. Calculate the electric potential at point Q.
3. Subtract the electric potential at P from the electric potential at Q to find the potential difference.

Slide 6: Example Calculation

Given:

Charge at P: +2 μC
Charge at Q: +4 μC
Distance between P and Q: 3 m Steps:

Calculate the electric potential at point P:
- Vp = k * (q / r)

Calculate the electric potential at point Q:
- Vq = k * (q / r)

Subtract the electric potential at P from the electric potential at Q to find the potential difference:
- ΔV = Vq - Vp Substitute the values and calculate ΔV.

Slide 7: Solution - Calculation of Electric Potential at P

Given information:
- Charge at P: +2 μC
- Distance between P and Q: 3 m
Using the equation for electric potential: Vp = k * (q / r) Substitute the values and calculate Vp.

Slide 8: Solution - Calculation of Electric Potential at Q

Given information:
- Charge at Q: +4 μC
- Distance between P and Q: 3 m
Using the equation for electric potential: Vq = k * (q / r) Substitute the values and calculate Vq.

Slide 9: Solution - Calculation of Potential Difference

Given information:
- Vp: Electric potential at P
- Vq: Electric potential at Q
Using the formula for potential difference: ΔV = Vq - Vp Substitute the calculated values of Vp and Vq, and calculate ΔV.

Slide 10: Conclusion

In this lecture, we covered the concept of potential due to different charge distributions and the calculation of potential difference between two points.
Understanding electric potential and potential difference is key to analyzing electric fields and charge behavior.
Practice calculating potential differences in various scenarios to strengthen your understanding.

Slide 11: Electric Potential Due to Continuous Charge Distribution

The concept of electric potential can also be applied to continuous charge distributions.
Instead of a point charge, we consider a continuous distribution of charge along a line, surface, or volume.
The principles and formulas for calculating potential difference remain the same for continuous distributions.

Slide 12: Potential Due to Line of Charge

Consider a linear charge distribution with charge density λ (charge per unit length) along a straight line.
The electric potential at a point P due to this line of charge is given by the equation: V = k * λ * ln(b/a)

In this equation:

V represents the electric potential at point P
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
λ is the charge density of the line of charge
a and b are the distances from the ends of the line of charge to the point P

Slide 13: Example - Electric Potential Due to Line of Charge

Let’s calculate the electric potential at a point P located at a distance r from a line of charge with a charge density λ.
Given:
- Charge density (λ): 4 μC/m
- Distance from line of charge to point P (r): 2 m
Using the equation: V = k * λ * ln(b/a)
Calculate the electric potential at point P.

Slide 14: Potential Due to Charged Ring

Consider a ring with charge Q uniformly distributed along its circumference.
The electric potential at a point on the axis of the ring due to this charged ring is given by the equation: V = k * (Q / √(x^2 + R^2))

In this equation:

V represents the electric potential at the point on the axis
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
Q is the total charge of the ring
x is the distance along the axis from the center of the ring
R is the radius of the ring

Slide 15: Example - Electric Potential Due to Charged Ring

Let’s calculate the electric potential on the axis of a charged ring at a distance x from the center of the ring.
Given:
- Total charge of the ring (Q): 6 μC
- Distance along the axis from the center of the ring (x): 1 m
- Radius of the ring (R): 2 m
Using the equation: V = k * (Q / √(x^2 + R^2))
Calculate the electric potential on the axis at distance x from the center.

Slide 16: Potential Due to Charged Disk

Consider a disk with charge density σ (charge per unit area) and radius R.
The electric potential at a point on the axis perpendicular to the disk due to this charged disk is given by the equation: V = (σ / 2ε₀) * (1 - √(1 + h^2 / R^2))

In this equation:

V represents the electric potential at the point on the axis
σ is the charge density of the disk
ε₀ is the permittivity of free space (8.85 x 10^-12 C^2/Nm^2)
h is the perpendicular distance from the disk to the point on the axis

Slide 17: Example - Electric Potential Due to Charged Disk

Let’s calculate the electric potential on the axis perpendicular to a charged disk at a distance h from the disk.
Given:
- Charge density (σ): 3 μC/m^2
- Distance from the disk to the point on the axis (h): 0.5 m
- Radius of the disk (R): 2 m
Using the equation: V = (σ / 2ε₀) * (1 - √(1 + h^2 / R^2))
Calculate the electric potential on the axis at distance h from the disk.

Slide 18: Potential Due to Spherical Shell

Consider a spherical shell with total charge Q uniformly distributed on its surface and radius R.
The electric potential at a point outside the spherical shell due to this charged shell is given by the equation: V = k * (Q / r)

In this equation:

V represents the electric potential at the point outside the shell
k is the electrostatic constant (9 x 10^9 Nm^2/C^2)
Q is the total charge of the shell
r is the distance from the center of the shell to the point outside the shell

Slide 19: Example - Electric Potential Due to Spherical Shell

Let’s calculate the electric potential at a point outside a charged spherical shell.
Given:
- Total charge of the shell (Q): 8 μC
- Distance from the center of the shell to the point outside (r): 5 m
Using the equation: V = k * (Q / r)
Calculate the electric potential at the point outside the shell.

Slide 20: Conclusion

In this section, we explored the calculation of electric potential due to various charge distributions.
We examined potential due to a line of charge, a charged ring, a charged disk, and a spherical shell.
Each distribution requires a specific equation to calculate the electric potential at a point.
Understanding these formulas and applying them correctly is essential for solving problems involving potential due to continuous charge distributions.

Slide 21: Examples of Potential Difference Calculations

Example 1:
- Given:
  - V1: Electric potential at point A = 10 V
  - V2: Electric potential at point B = 5 V
- Calculate the potential difference (ΔV) between points A and B.
Example 2:
- Given:
  - V1: Electric potential at point X = -8 V
  - V2: Electric potential at point Y = 2 V
- Calculate the potential difference (ΔV) between points X and Y.
Example 3:
- Given:
  - V1: Electric potential at point M = 20 V
  - V2: Electric potential at point N = 25 V
- Calculate the potential difference (ΔV) between points M and N.
Example 4:
- Given:
  - V1: Electric potential at point P = 0 V
  - V2: Electric potential at point Q = -15 V
- Calculate the potential difference (ΔV) between points P and Q.
Example 5:
- Given:
  - V1: Electric potential at point E = 12 V
  - V2: Electric potential at point F = 12 V
- Calculate the potential difference (ΔV) between points E and F.

Slide 22: Potential Difference Calculation Formula

The formula to calculate the potential difference (ΔV) between two points is:
- ΔV = V2 - V1
Remember that ΔV represents the difference in electric potential between the second point (V2) and the first point (V1).

Slide 23: Potential Due to Continuous Line Charge - Derivation

Consider a continuous line charge with charge density λ (charge per unit length).
To determine the potential difference between two points along the line, we integrate the potential contribution of each infinitesimally small charge element.
The derivation involves using the formula for electric potential due to a point charge.
The result is the formula for the potential at a point on the line due to the entire line charge distribution.
The final formula is:
- V = λ * ln(b/a) / (4πε₀)
  - ln: natural logarithm
  - a, b: distances from the ends of the line charge to the point of interest
  - ε₀: permittivity of vacuum

Slide 24: Potential Due to Continuous Line Charge - Example Calculation

Given:
- Charge density (λ) of the line charge = 2 μC/m
- Distance from one end (a) = 1 m
- Distance from the other end (b) = 3 m
- ε₀: permittivity of vacuum = 8.85 x 10^-12 C^2/Nm^2
Using the formula: V = λ * ln(b/a) / (4πε₀)
Calculate the electric potential at a point on the line charge.

Slide 25: Potential Due to Continuous Surface Charge - Derivation

Consider a continuous surface charge with charge density σ (charge per unit area).
To determine the potential difference between two points above the surface, we integrate the potential contribution of each infinitesimally small charge element.
The derivation involves using the formula for electric potential due to a point charge.
The result is the formula for the potential above the surface due to the entire surface charge distribution.
The final formula is:
- V = σ * (1/4ε₀) * ∫ [(1 / √(x^2 + y^2)) * dy]
  - x: distance along the axis perpendicular to the surface
  - y: distance along the surface
  - σ: surface charge density
  - ε₀: permittivity of vacuum
  - ∫: integral symbol

Slide 26: Potential Due to Continuous Surface Charge - Example Calculation

Given:
- Surface charge density (σ) = 3 μC/m^2
- Distance along the axis perpendicular to the surface (x) = 2 m
- ε₀: permittivity of vacuum = 8.85 x 10^-12 C^2/Nm^2
Using the formula: V = σ * (1/4ε₀) * ∫ [(1 / √(x^2 + y^2)) * dy]
Calculate the electric potential at a point on the axis above the surface charge.

Slide 27: Potential Due to Continuous Volume Charge - Derivation

Consider a continuous volume charge with charge density ρ (charge per unit volume).
To determine the potential difference between two points inside the volume, we integrate the potential contribution of each infinitesimally small charge element.
The derivation involves using the formula for electric potential due to a point charge.
The result is the formula for the potential inside the volume due to the entire volume charge distribution.
The final formula is:
- V = (1/4πε₀) * ∫ [(ρ / r) * dV]
  - r: distance from the charge element to the point inside the volume
  - ρ: volume charge density
  - ε₀: permittivity of vacuum
  - ∫: integral symbol
  - dV: infinitesimally small volume element

Slide 28: Potential Due to Continuous Volume Charge - Example Calculation

Given:
- Volume charge density (ρ) = 5 μC/m^3
- Distance from the center of the volume charge (r) = 4 m
- ε₀: permittivity of vacuum = 8.85 x 10^-12 C^2/Nm^2
Using the formula: V = (1/4πε₀) * ∫ [(ρ / r) * dV]
Calculate the electric potential at a point inside the volume charge.

Slide 29: Conclusion

In this lecture, we explored examples and formulas for calculating potential difference between two points.
We covered potential difference due to continuous line charges, surface charges, and volume charges.
The derivations involved integrals and the application of relevant charge density values.
Practice applying the formulas to various scenarios to improve your understanding and problem-solving skills.

Slide 30: Q&A Session

This concludes our lecture on potential difference calculations.
Now, let’s move on to the Q&A session.
Please feel free to ask any questions you may have regarding the topic.
We will discuss and clarify any doubts or concerns you may have.
Thank you for your attention!