What is the electric potential at the point indicated with the dot in FIGURE EX25.31?

Verified step by step guidance

1. Step 1: Recall the formula for electric potential due to a point charge: \( V = \frac{k \cdot q}{r} \), where \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \)), \( q \) is the charge, and \( r \) is the distance from the charge to the point of interest.
2. Step 2: Identify the charges and their distances from the point indicated by the dot. The charges are \( +2.0 \, \text{nC} \) (top and left) and \( -1.0 \, \text{nC} \) (bottom right). The distances are \( 4.0 \, \text{cm} \), \( 3.0 \, \text{cm} \), and the diagonal distance to the negative charge can be calculated using the Pythagorean theorem: \( r = \sqrt{(3.0 \, \text{cm})^2 + (4.0 \, \text{cm})^2} \).
3. Step 3: Convert all distances from centimeters to meters for consistency in SI units. For example, \( 4.0 \, \text{cm} = 0.04 \, \text{m} \), \( 3.0 \, \text{cm} = 0.03 \, \text{m} \), and the diagonal distance \( r = \sqrt{(0.03)^2 + (0.04)^2} \).
4. Step 4: Calculate the electric potential at the dot due to each charge separately using \( V = \frac{k \cdot q}{r} \). For the top charge, use \( q = +2.0 \, \text{nC} \) and \( r = 0.04 \, \text{m} \). For the left charge, use \( q = +2.0 \, \text{nC} \) and \( r = 0.03 \, \text{m} \). For the bottom right charge, use \( q = -1.0 \, \text{nC} \) and the diagonal distance calculated earlier.
5. Step 5: Add the potentials from all three charges algebraically (taking into account the sign of each charge) to find the total electric potential at the dot. Remember that electric potential is a scalar quantity, so you simply sum the contributions.

Key Concepts

Electric Potential

Electric potential, measured in volts, is the amount of electric potential energy per unit charge at a point in an electric field. It indicates how much work would be done to move a positive test charge from a reference point to the specified point without any acceleration. The electric potential due to a point charge can be calculated using the formula V = k * (q/r), where V is the electric potential, k is Coulomb's constant, q is the charge, and r is the distance from the charge to the point of interest.

Superposition Principle

The superposition principle states that the total electric potential at a point due to multiple charges is the algebraic sum of the potentials due to each charge individually. This means that when calculating the electric potential at a specific point, one can calculate the contribution from each charge separately and then sum these contributions to find the total potential at that point. This principle is fundamental in electrostatics and simplifies the analysis of systems with multiple charges.

Coulomb's Law

Coulomb's Law describes the force between two point charges and is essential for understanding electric interactions. It states that the electric force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law helps in determining the electric field created by a charge, which is crucial for calculating the electric potential at a point in the field created by multiple charges.