For Arrangement 2, determine the following: The electrostatic potential at the center of the square.
Understand the Problem
The question is asking to determine the electrostatic potential at the center of a square configuration of charges given in Arrangement 2. It involves applying the formula for electric potential and the contributions from multiple charges arranged in a specific geometric configuration.
Answer
The electrostatic potential at the center of the square is $V = 0$.
Answer for screen readers
The electrostatic potential at the center of the square is $V_{total} = 0$.
Steps to Solve
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Identify the Charges and Configuration
In Arrangement 2, we have four charges: two positive charges (+2Q) at the top left and bottom left corners, and two negative charges (-2Q) at the top right and bottom right corners of the square. -
Calculate the Distance from Center to Each Charge
The distance from the center of the square to each corner is equal to half the length of the diagonal. For a square of side length (s), the distance (r) is given by:
$$ r = \frac{s \sqrt{2}}{2} $$ -
Calculate the Electric Potential from Each Charge
The electric potential (V) due to a point charge is calculated using the formula:
$$ V = k \frac{Q}{r} $$ Here, each charge contributes to the total potential at the center. -
Determine the Contribution of Each Charge to the Total Potential
Since there are two +2Q charges and two -2Q charges, their contributions are:
- For +2Q charges:
$$ V_{+} = 2 \cdot k \frac{2Q}{r} $$ - For -2Q charges:
$$ V_{-} = 2 \cdot k \frac{-2Q}{r} $$
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Combine the Potentials
The total potential at the center (V_{total}) is the sum of the contributions:
$$ V_{total} = V_{+} + V_{-} $$
Substituting the values:
$$ V_{total} = 2 \cdot k \frac{2Q}{r} + 2 \cdot k \frac{-2Q}{r} $$ -
Simplify the Expression
When you simplify, notice that the positives and negatives cancel out:
$$ V_{total} = 0 $$
The electrostatic potential at the center of the square is $V_{total} = 0$.
More Information
In this arrangement, the contributions to the electric potential from the positive and negative charges perfectly cancel each other out, resulting in zero potential at the center of the square.
Tips
- Forgetting to consider the signs of the charges when calculating potential contributions.
- Confusing the total distance (diagonal) with the individual distances to each charge.
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