Potential Due To Different Charge Distributions

Calculation of potential due to a dipole at a general point in space

The potential due to a dipole is given by the equation:
- V = k(q/r1 - q/r2)
- where k = 1/4πε₀ and ε₀ is the permittivity of free space.
Here, q is the magnitude of the charge separated by a distance d and r1 and r2 are distances from the positive and negative charges to the point where potential is to be calculated.
Recall that the electric potential at a point is the amount of work done in bringing a unit positive charge from infinity to that point.
In this lecture, we will focus on calculating the potential at a general point due to a dipole.

Calculation of Potential Due to a Dipole

Consider a dipole with charges +q and -q, separated by a distance d.
Let’s assume that the positive charge is at the origin (0, 0, 0) and the negative charge is at (0, 0, d).
We want to calculate the potential at a general point P(x, y, z) due to this dipole.
To do this, we will calculate the potential due to the positive charge and the potential due to the negative charge separately, and then subtract them.

Calculating Potential Due to Positive Charge

To calculate the potential due to the positive charge at P(x, y, z), we use the equation:
- V₁ = k(q/r₁)
- where r₁ is the distance between the positive charge and the point P.
The distance r₁ can be calculated using the Pythagorean theorem:
- r₁ = √(x² + y² + z²)
Plugging in the values, we get:
- V₁ = k(q/√(x² + y² + z²))

Calculating Potential Due to Negative Charge

To calculate the potential due to the negative charge at P(x, y, z), we use the equation:
- V₂ = k(-q/r₂)
- where r₂ is the distance between the negative charge and the point P.
The distance r₂ can be calculated using the Pythagorean theorem:
- r₂ = √(x² + y² + (d-z)²)
Plugging in the values, we get:
- V₂ = k(-q/√(x² + y² + (d-z)²))

Total Potential Due to a Dipole

The total potential at point P(x, y, z) due to the dipole is given by:
- V = V₁ - V₂
Substituting the values of V₁ and V₂:
- V = k(q/√(x² + y² + z²)) - k(-q/√(x² + y² + (d-z)²))
Simplifying the equation further, we get:
- V = kq[1/√(x² + y² + z²) + 1/√(x² + y² + (d-z)²)]

Example 1

Let’s consider a dipole with charges +2e and -2e, separated by a distance of 0.1 m.
We want to find the potential at a point P(0.2, 0.3, 0.4) m due to this dipole.
Using the formula derived earlier, we can calculate the potential as:
- V = kq[1/√(x² + y² + z²) + 1/√(x² + y² + (d-z)²)]
Plugging in the values, we get:
- V = (1/4πε₀)(2e)[1/√(0.2² + 0.3² + 0.4²) + 1/√(0.2² + 0.3² + (0.1-0.4)²)]

Calculation of Potential at Point P

Evaluating the expression further, we substitute the constant values:
- V = (9 × 10^9 Nm²/C²)(2 × 1.6 × 10^-19 C)[1/√(0.2² + 0.3² + 0.4²) + 1/√(0.2² + 0.3² + (0.1-0.4)²)]
Performing the calculations, we get:
- V ≈ 1.1 × 10^9 V
Hence, the potential at point P(0.2, 0.3, 0.4) due to the given dipole is approximately 1.1 × 10^9 volts.

Calculation of Potential Due to a Dipole - Summary

The potential at a general point due to a dipole can be calculated using the equation:
- V = k(q/r1 - q/r2)
By considering the positive and negative charges separately, we can derive expressions for the potential at a general point due to the positive charge (V₁) and the negative charge (V₂).
The total potential at the point is obtained by subtracting the potential due to the negative charge from the potential due to the positive charge.
Calculations involve plugging the given values into the equations and simplifying the expressions accordingly.

Key Takeaways

The potential at a point due to a dipole is given by the equation V = k(q/r1 - q/r2), where k = 1/4πε₀.
The potential due to the positive charge can be calculated using V₁ = k(q/r₁), where r₁ is the distance between the positive charge and the point.
The potential due to the negative charge can be calculated using V₂ = k(-q/r₂), where r₂ is the distance between the negative charge and the point.
The total potential is obtained by subtracting V₂ from V₁.
Remember to plug in the given values and simplify the expressions to calculate the final potential at a general point.

Summary

In this lecture, we learned about calculating the potential at a general point due to a dipole.
We derived the formulas for potential due to the positive and negative charges separately, and then subtracted them to find the total potential at a point.
A practical example was solved to illustrate the calculation process.
Understanding and applying these concepts is essential for solving problems related to potential due to different charge distributions.

Electric Potential Due to a Uniformly Charged Sphere

A uniformly charged sphere with radius R and total charge Q can be used to calculate the electric potential at a point outside the sphere.
The equation for the electric potential due to a uniformly charged sphere at a point outside the sphere is given by:
- V = (kQ) / (r)
where k = 1/4πε₀ and ε₀ is the permittivity of free space, Q is the charge of the sphere, and r is the distance between the center of the sphere and the point where the potential is to be calculated.
This equation assumes that the sphere is a solid, non-conducting sphere with a constant charge density.

Electric Potential Due to a Uniformly Charged Sphere (Contd.)

The electric potential inside the uniformly charged sphere is constant and equal to the potential at the surface of the sphere.
The equation for the electric potential inside the uniformly charged sphere is given by:
- V = (kQ) / (R)
where R is the radius of the sphere.
The potential outside the sphere decreases inversely with the distance from the center of the sphere, while the potential inside the sphere remains constant.

Example: Electric Potential Due to a Uniformly Charged Sphere

Let’s consider a uniformly charged sphere with a radius of 0.5 m and a charge of 10 μC.
We want to find the electric potential at a point located 1 m away from the center of the sphere.
Using the formula for electric potential due to a uniformly charged sphere at a point outside the sphere, we have:
- V = (kQ) / (r)
- V = (9 × 10^9 Nm²/C²)(10 × 10^-6 C) / (1 m)
Calculating the value, we get:
- V = 9 × 10^3 V
Hence, the electric potential at the given point is 9,000 volts.

Electric Potential Due to a Point Charge

The electric potential due to a point charge at a distance r from the charge can be calculated using the equation:
- V = (kQ) / (r)
where k = 1/4πε₀, Q is the charge, and r is the distance between the point charge and the point where the potential is to be calculated.
The electric potential decreases with increasing distance from the point charge.
The electric potential at infinity is zero, as it represents the reference point with zero potential energy.

Electric Potential Due to Multiple Point Charges

The electric potential at a point due to multiple point charges is the algebraic sum of the individual potentials due to each charge.
To calculate the electric potential at a point due to multiple point charges, use the equation:
- V = V₁ + V₂ + V₃ + …
where V₁, V₂, V₃, … are the individual potentials due to each point charge.
Make sure to include the signs of each potential, as potential is a scalar quantity.
This calculation applies to both point charges with the same and opposite signs.

Electric Potential Due to Multiple Point Charges (Contd.)

If the point charges have opposite signs and are close together, the potential near each charge is large compared to the potential far away.
In such cases, it is important to take into account the distance and the sign of each charge while calculating the potential.
Remember that the potential is a scalar quantity and can be either positive or negative depending on the sign of the charges and the observer’s reference point.
Electric potential is always positive for positive charges and negative for negative charges.

Example: Electric Potential Due to Multiple Point Charges

Let’s consider two point charges: +5 μC located at position A(0, 0, 0) and -2 μC located at position B(0, 0, 2 m).
We want to find the electric potential at a point P(1, 1, 1) m due to these charges.
First, calculate the electric potentials due to each point charge at point P using the equation V = (kQ) / (r).
Then, add the potentials together to find the total potential at point P.
Take into account the signs and distances of the charges while calculating the potentials.

Example: Electric Potential Due to Multiple Point Charges (Contd.)

The electric potential due to the +5 μC charge at point P(1, 1, 1) can be calculated as:
- V₁ = (kQ₁) / (r₁)
- V₁ = (9 × 10^9 Nm²/C²)(5 × 10^-6 C) / (√(1² + 1² + 1²) m)
The electric potential due to the -2 μC charge at point P(1, 1, 1) can be calculated as:
- V₂ = (kQ₂) / (r₂)
- V₂ = (9 × 10^9 Nm²/C²)(-2 × 10^-6 C) / (√(1² + 1² + (1-2)²) m)
Finally, add the two potentials V₁ and V₂ to find the total potential V at point P.

Example: Electric Potential Due to Multiple Point Charges (Contd.)

Evaluating the equations for V₁ and V₂ at point P(1, 1, 1) and performing the calculations, we get:
- V₁ ≈ 4.1 × 10^5 V
- V₂ ≈ -3.3 × 10^9 V
Adding the potentials V₁ and V₂, we get the total potential V:
- V = V₁ + V₂
- V ≈ 3.3 × 10^9 V
Hence, the electric potential at point P(1, 1, 1) due to the given point charges is approximately 3.3 × 10^9 volts.

Summary

The electric potential due to a uniformly charged sphere outside the sphere is given by the equation V = (kQ) / (r), where k = 1/4πε₀, Q is the charge, and r is the distance from the center of the sphere.
Inside the sphere, the potential is constant and equal to the potential at the surface of the sphere.
The electric potential due to a point charge is given by V = (kQ) / (r), where k = 1/4πε₀, Q is the charge, and r is the distance from the charge.
The electric potential due to multiple point charges is the algebraic sum of the individual potentials due to each charge.
Make sure to include the signs of each potential when calculating the total potential.

Potential Due To Different Charge Distributions

Calculation of potential due to a dipole at a general point in space

The potential due to a dipole is given by the equation: V = k(q/r1 - q/r2), where k = 1/4πε₀ and ε₀ is the permittivity of free space.

The potential due to the positive charge can be calculated using V₁ = k(q/r₁), where r₁ is the distance between the positive charge and the point.

The potential due to the negative charge can be calculated using V₂ = k(-q/r₂), where r₂ is the distance between the negative charge and the point.

The total potential is obtained by subtracting V₂ from V₁.

Calculations involve plugging the given values into the equations and simplifying the expressions accordingly.

Electric Potential Due to a Uniformly Charged Sphere

The electric potential due to a uniformly charged sphere at a point outside the sphere is given by V = (kQ) / (r).
The electric potential inside the uniformly charged sphere is constant and equal to the potential at the surface of the sphere, given by V = (kQ) / (R).
The potential outside the sphere decreases inversely with the distance from the center of the sphere.
The potential at infinity is zero, representing the reference point with zero potential energy.
The potential at a point inside the sphere is constant and equal to the potential at the surface of the sphere.

Example: Electric Potential Due to a Uniformly Charged Sphere

Consider a uniformly charged sphere with radius R = 0.5 m and charge Q = 10 μC.
Calculate the electric potential at a point located 1 m away from the center of the sphere.
Use the formula V = (kQ) / (r), where k = 1/4πε₀ and r is the distance from the center of the sphere.
Plugging in the values, we get V = (9 × 10^9 Nm²/C²)(10 × 10^-6 C) / (1 m).
Solving the equation, the potential is found to be V ≈ 9 × 10^3 V.

Electric Potential Due to a Point Charge

The electric potential due to a point charge at a distance r from the charge is given by V = (kQ) / (r).
The potential decreases with increasing distance from the point charge.
The potential at infinity is zero, representing the reference point with zero potential energy.
The potential is positive for positive charges and negative for negative charges.
The difference in potential between two points represents the work done in moving a unit positive charge from one point to another.

Electric Potential Due to Multiple Point Charges

The electric potential at a point due to multiple point charges is the algebraic sum of the individual potentials due to each charge.
Calculate the electric potential at a point using the equation V = V₁ + V₂ + V₃ + …, where V₁, V₂, V₃, … are the individual potentials due to each point charge.
Consider the signs of each potential, as potential is a scalar quantity.
Positive potentials correspond to regions of high potential energy, while negative potentials correspond to regions of low potential energy.
Take into account the signs and distances of the charges while calculating the potentials.

Example: Electric Potential Due to Multiple Point Charges

Two point charges are present: +5 μC at position A(0, 0, 0) and -2 μC at position B(0, 0, 2 m).
Calculate the electric potential at a point P(1, 1, 1) m due to these charges.
Use the equation V = (kQ) / (r) to calculate the potentials due to each charge.
The potential due to the +5 μC charge can be calculated as V₁ = (9 × 10^9 Nm²/C²)(5 × 10^-6 C) / (√(1² + 1² + 1²) m).
The potential due to the -2 μC charge can be calculated as V₂ = (9 × 10^9 Nm²/C²)(-2 × 10^-6