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To solve this problem, we need to calculate the total electric potential at point P, which is at the center of the rectangle formed by the six charges.

The formula for the electric potential \( V \) due to a point charge \( q \) at a distance \( r \) from the charge is given by:
\[
V = \frac{kq}{r}
\]
where:
- \( V \) is the electric potential,
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \)),
- \( q \) is the charge,
- \( r \) is the distance from the charge to the point where the potential is being calculated.

### Step 1: Understanding the Charge Configuration
Given the diagram:
- The six charges are arranged in a rectangular configuration.
- The charges are positioned as follows:
  - \( +7q \) (top-left corner),
  - \( +3q \) (top-center),
  - \( +5q \) (top-right),
  - \( -5q \) (bottom-left),
  - \( -3q \) (bottom-center),
  - \( +7q \) (bottom-right).

### Step 2: Symmetry Considerations
Since potential is a scalar quantity, we can sum up the potentials from all the charges without worrying about direction. The potential due to each charge will be calculated at the distance from point P, which is at the center.

We can divide the charges into symmetric pairs to simplify:
1. The two charges \( +3q \) (top-center) and \( -3q \) (bottom-center) are equidistant from point P and will have opposite contributions, so their total contribution to the potential is 0.
2. Similarly, the charges \( +7q \) (top-left) and \( +7q \) (bottom-right) will have the same distance from P and contribute equally to the total potential.
3. The charges \( +5q \) (top-right) and \( -5q \) (bottom-left) will also cancel out some of their potential contributions based on their magnitudes.

### Step 3: Distance Calculation
The charges are arranged in a rectangle, so we need to calculate the distance from each charge to point P.

- For the charges directly across from the center (like \( +3q \) and \( -3q \)), the distance to P is half the height or width of the rectangle (i.e., \( d/2 \)).
- For diagonal charges (like \( +7q \) in the corners), the distance is the diagonal of the rectangle, which can be calculated using the Pythagorean theorem:
\[
r_{\text{diagonal}} = \sqrt{\left(\frac{d}{2}\right)^2 + \left(\frac{d}{2}\right)^2} = \frac{d\sqrt{2}}{2}
\]

### Step 4: Total Potential Calculation
Now, we'll calculate the total potential at point P by summing up the potentials from all six charges.

Let me calculate this.The total electric potential at location \( P \), which is at the center of the rectangle, is approximately \( 747,130.77 \, \text{V} \) (Volts).

Would you like further details or explanations on any part of this calculation?

Here are some related questions:
1. How does electric potential differ from electric field at a point?
2. What happens if we change the charge magnitudes or positions slightly?
3. How would the total electric potential change if the rectangle were a square?
4. Can you explain how the potential would be affected if the center-aligned charges had different values?
5. What role does symmetry play in simplifying electrostatic calculations?

**Tip:** When calculating electric potential, remember that it's a scalar quantity, so we simply sum up the potentials without worrying about direction. This can often simplify problems involving multiple charges.

The drawing shows six point charges arranged in a rectangle. The value of q is 2.04 μC, and the distance d is 0.486 m. Find the total electric potential at location P, which is at the center of the rectangle.

This problem explores the electric potential at a central point of a rectangle formed by six charges. It involves calculating the contributions from each charge using Coulomb's law and applying symmetry considerations. Suitable for undergraduate students studying electrostatics.

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