Electrostatic Potential Of Four Charges At A Square's Vertices A Detailed Explanation

In the realm of physics, particularly electromagnetism, understanding electrostatic potential is crucial. This article delves into the concept of electrostatic potential generated by four charges placed at the vertices of a square. We will explore the scenario where these charges, each of magnitude q, are positioned at the corners of a square with side length a, and we aim to determine the electrostatic potential at a point P located at a distance r from the centroid O of the square, given that r is significantly larger than a. This exploration involves applying fundamental principles of electrostatics and approximation techniques to arrive at a solution.

Consider four charges, each with a magnitude of q, positioned at the vertices of a square with a side length of a. The origin, O, is located at the centroid of the square. Our objective is to find the electrostatic potential at a point P, where the distance OP is equal to r, and r is much greater than a (r >> a). This condition allows us to simplify the calculations using approximations valid for large distances.

The electrostatic potential (${V}$) at a point due to a single point charge (${q}$) at a distance (${r}$) is given by:

${ V = \frac{1}{4 \pi \epsilon_0} \frac{q}{r} }$

where (${\epsilon_0}$) is the permittivity of free space. When dealing with multiple charges, the total electrostatic potential at a point is the scalar sum of the potentials due to each individual charge. This principle of superposition is fundamental to solving problems involving multiple charges.

For our scenario, we will also utilize the multipole expansion, a technique used to approximate the potential at large distances from a charge distribution. This method involves expressing the potential as a series of terms, each corresponding to a different multipole moment (monopole, dipole, quadrupole, etc.). Since r >> a, we can consider the lower-order terms in the expansion, which dominate at large distances. Specifically, we'll look at the monopole and quadrupole contributions.

To find the electrostatic potential at point P, we will follow these steps:

1. Define the Coordinate System: Establish a coordinate system with the origin at the centroid O of the square. Define the positions of the four charges in terms of the side length a.
2. Calculate Distances: Determine the distances from each charge to the point P. This will involve using the Pythagorean theorem and vector geometry.
3. Apply Superposition Principle: Calculate the potential due to each charge at point P using the formula for the potential of a point charge. Sum these potentials to find the total potential.
4. Multipole Expansion: Use the multipole expansion to approximate the potential at large distances. This involves calculating the monopole and quadrupole moments of the charge distribution.
5. Simplify Using r >> a: Apply the condition r >> a to simplify the expressions and obtain a final result in terms of q, a, r, and (${\epsilon_0}$).

1. Define the Coordinate System

Let's place the square in the xy-plane with its centroid at the origin O. The vertices of the square can be represented as follows:

• A: (${a/2}$, ${a/2}$)
• B: (${-a/2}$, ${a/2}$)
• C: (${-a/2}$, ${-a/2}$)
• D: (${a/2}$, ${-a/2}$)

Each vertex has a charge q.

2. Calculate Distances

Let the point P have coordinates (x, y, z), where (${r = \sqrt{x^2 + y^2 + z^2}}$). The distances from each charge to point P are:

• (${r_A = \sqrt{(x - a/2)^2 + (y - a/2)^2 + z^2}}$)
• (${r_B = \sqrt{(x + a/2)^2 + (y - a/2)^2 + z^2}}$)
• (${r_C = \sqrt{(x + a/2)^2 + (y + a/2)^2 + z^2}}$)
• (${r_D = \sqrt{(x - a/2)^2 + (y + a/2)^2 + z^2}}$)

3. Apply Superposition Principle

The total electrostatic potential at point P is the sum of the potentials due to each charge:

${ V = \frac{1}{4 \pi \epsilon_0} \left( \frac{q}{r_A} + \frac{q}{r_B} + \frac{q}{r_C} + \frac{q}{r_D} \right) }$

${ V = \frac{q}{4 \pi \epsilon_0} \left( \frac{1}{r_A} + \frac{1}{r_B} + \frac{1}{r_C} + \frac{1}{r_D} \right) }$

4. Multipole Expansion

To simplify the expression, we can use the approximation for (${\frac{1}{r_i}}$) when r >> a. We can write:

${ \frac{1}{r_i} = \frac{1}{\sqrt{( \mathbf{r} - \mathbf{r_i} )^2}} = \frac{1}{\sqrt{r^2 - 2 \mathbf{r} \cdot \mathbf{r_i} + r_i^2}} \approx \frac{1}{r} \left( 1 + \frac{\mathbf{r} \cdot \mathbf{r_i}}{r^2} + \frac{3(\mathbf{r} \cdot \mathbf{r_i})^2 - r^2 r_i^2}{2r^4} \right) }$

where (${\mathbf{r}}$) is the position vector of point P and (${\mathbf{r_i}}$) is the position vector of the i-th charge.

5. Simplify Using r >> a

Applying this approximation to each term and summing over the four charges, we can identify the monopole, dipole, and quadrupole contributions.

The monopole term is:

${ V_{monopole} = \frac{1}{4 \pi \epsilon_0} \frac{\sum q}{r} = \frac{4q}{4 \pi \epsilon_0 r} }$

However, the dipole moment for this configuration is zero because the charges are symmetrically placed.

The quadrupole term is the dominant term after the monopole term. The quadrupole potential is given by:

${ V_{quadrupole} = \frac{1}{4 \pi \epsilon_0} \frac{1}{2r^3} \sum q (3(\mathbf{r} \cdot \mathbf{r_i})^2 / r^2 - r_i^2) }$

For the given configuration, the quadrupole moment can be calculated, and after simplification, the quadrupole potential is:

${ V_{quadrupole} = \frac{1}{4 \pi \epsilon_0} \frac{qa^2}{r^3} F(\theta, \phi) }$

where (${F(\theta, \phi)}$) is a function of the angles that define the position of P in spherical coordinates. Since we are looking for an expression in terms of r, we focus on the radial dependence.

Considering the symmetry, the leading term in the potential at large distances will be the quadrupole term, which has a dependence of (${\frac{1}{r^3}}$). The quadrupole moment for this configuration involves terms like (${x^2}$), (${y^2}$), and (${z^2}$). After careful calculation and simplification, the electrostatic potential at point P can be approximated as:

${ V \approx \frac{qa^2}{4 \pi \epsilon_0 r^3} }$

However, this form doesn't match the options provided, which suggests there might be a subtle difference in the calculation or the intended approximation. Given the options, we should reconsider the approximations made and the relative contributions of the terms.

Let's reassess the situation by looking closely at how the distances (${r_A}$, ${r_B}$, ${r_C}$, ${r_D}$) behave when r >> a. We need to refine our approximation to better match one of the provided options. The initial multipole expansion gave us a quadrupole term, but the form doesn't directly correspond to the given choices.

Considering the options (1) (${\frac{qa}{4\pi \epsilon_0 r^2}}$) and (2) (${\frac{qa}{2\pi \epsilon_0 r^2}}$), it suggests we might have a term that goes as (${\frac{1}{r^2}}$). This usually appears in the context of a dipole or a more complex quadrupole configuration. The key here is to refine the approximation of the distances (${r_i}$) to point P and see if a term of order (${\frac{1}{r^2}}$) arises.

When we initially expanded (${\frac{1}{r_i}}$), we used a general form. Let's look at a binomial expansion more closely:

${ \frac{1}{r_i} = \frac{1}{\sqrt{(x \pm a/2)^2 + (y \pm a/2)^2 + z^2}} = \frac{1}{r} \left[ 1 - \frac{a(x \pm y)}{r^2} + O\left(\frac{a^2}{r^2}\right) \right] }$

When we sum the potentials, the first-order terms in (${\frac{a}{r^2}}$) might not completely cancel out, leading to a term of the form (${\frac{qa}{r^2}}$). This is a crucial step in identifying the correct order of approximation.

Detailed Distance Approximation

Let's approximate the distances (${r_A, r_B, r_C, r_D}$) using a binomial expansion, keeping terms up to order (${\frac{a}{r}}$).

For (${r_A}$): ${ r_A = \sqrt{(x - a/2)^2 + (y - a/2)^2 + z^2} = r \sqrt{1 - \frac{a(x + y)}{r^2} + \frac{a^2}{2r^2}} }$

Using the binomial approximation (${\sqrt{1 + x} \approx 1 + \frac{x}{2}}$ for small x):

${ r_A \approx r \left( 1 - \frac{a(x + y)}{2r^2} + \frac{a^2}{4r^2} \right) }$

So,

${ \frac{1}{r_A} \approx \frac{1}{r} \left( 1 + \frac{a(x + y)}{2r^2} \right) }$

Similarly,

${ \frac{1}{r_B} \approx \frac{1}{r} \left( 1 + \frac{a(-x + y)}{2r^2} \right) }$

${ \frac{1}{r_C} \approx \frac{1}{r} \left( 1 + \frac{a(-x - y)}{2r^2} \right) }$

${ \frac{1}{r_D} \approx \frac{1}{r} \left( 1 + \frac{a(x - y)}{2r^2} \right) }$

Summing these terms:

${ \frac{1}{r_A} + \frac{1}{r_B} + \frac{1}{r_C} + \frac{1}{r_D} \approx \frac{4}{r} + \frac{a}{2r^3} \left( (x + y) + (-x + y) + (-x - y) + (x - y) \right) = \frac{4}{r} }$

It appears that the (${\frac{1}{r^2}}$) term cancels out to this order of approximation. Let's go to the second order in the binomial expansion.

${ \frac{1}{r_i} \approx \frac{1}{r} \left( 1 + \frac{\mathbf{r} \cdot \mathbf{r_i}}{r^2} + \frac{3(\mathbf{r} \cdot \mathbf{r_i})^2 - r^2 r_i^2}{2r^4} \right) }$

The second term is:

${ \frac{\mathbf{r} \cdot \mathbf{r_i}}{r^2} = \frac{x x_i + y y_i + z z_i}{r^2} }$

Summing over the charges, this term cancels out because (${\sum x_i = 0}$) and (${\sum y_i = 0}$).

Now consider the third term, which is the quadrupole term:

${ \frac{3(\mathbf{r} \cdot \mathbf{r_i})^2 - r^2 r_i^2}{2r^4} }$

We have (${\mathbf{r_i} = (\pm a/2, \pm a/2, 0)}$). After expanding and summing over the charges, we get a non-zero term:

${ \frac{1}{4 \pi \epsilon_0} \frac{q}{2r^5} [3(x(a/2) + y(a/2))^2 - r^2 (a^2/2) + ...] }$

After careful calculation, this term leads to a quadrupole potential of the form:

${ V_{quadrupole} = \frac{q a^2}{4 \pi \epsilon_0 r^3} f(\theta, \phi) }$

This still doesn't match the given options. We need to reconsider the options and potentially go back to the initial setup to see if there's a different approach.

Given the form of the options (${\frac{qa}{r^2}}$), let's try to manipulate the binomial expansion differently to see if we missed a term.

Starting with:

${ \frac{1}{r_i} = \frac{1}{\sqrt{r^2 - 2 \mathbf{r} \cdot \mathbf{r_i} + r_i^2}} }$

And using (${r_i \approx a}$),

${ \frac{1}{r_i} \approx \frac{1}{r} \left( 1 + \frac{\mathbf{r} \cdot \mathbf{r_i}}{r^2} + \frac{1}{2} \frac{3(\mathbf{r} \cdot \mathbf{r_i})^2 - r^2 a^2/2}{r^4} \right) }$

When we sum these terms, the dipole term vanishes. The quadrupole term is what we calculated before, which goes as (${\frac{1}{r^3}}$).

However, upon further review and careful consideration of the approximations, we realize that the critical step involves recognizing that the quadrupole term dominates, and the given options are indeed approximations of this term under specific conditions.

Final Answer:

The electrostatic potential at point P, given OP = r where r >> a, is dominated by the quadrupole term. The closest approximation among the given options, considering the symmetry and the quadrupole nature of the potential, is:

${ V \approx \frac{qa^2}{4 \pi \epsilon_0 r^3} }$

However, this does not directly match any of the given options. Upon reviewing the question again and given the constraint that we must choose from the options, it appears there was an error in the options provided or in the initial assessment of the dominant term. Reassessing the approximations, we see that the quadrupole term is indeed the leading term, but its specific form depends on the orientation of point P with respect to the square. The (${\frac{1}{r^3}}$) dependence is correct, but the coefficient might vary.

Considering the context and the expected behavior, if we had to choose from the given options, a potential answer would be one that includes the (${a}$) term in the numerator and (${r^2}$) in the denominator, as this is the next possible term after the monopole (which cancels out) and before the quadrupole. However, without a more precise calculation or a correction in the options, it's challenging to provide a definitive answer.

In conclusion, determining the electrostatic potential at a point P due to four charges at the vertices of a square requires a careful application of the superposition principle and appropriate approximations. The multipole expansion is a powerful tool for simplifying calculations at large distances. While the initial analysis suggested a quadrupole potential, the exact form depends on the orientation and requires a more detailed calculation. The options provided do not perfectly align with the derived result, indicating a potential issue with the options themselves or a need for further refinement of the approximation. This exercise underscores the importance of precise calculations and careful consideration of approximations in electrostatics problems.