We may define electrostatic potential at a point in an electrostatic field as the amount of work done in moving a unit positive test charge from infinity to that point against the electrostatic forces, along any path. Due to a single charge `q` , potential at a point distant `r` from the charge is `V = (q)/(4pi in_(0)r)`. The potential can be positive or negative. However, it is scalar quantity. The total amount of work done in bringing various charges to their respective postions from infinelty large mutual separations gives us the electric potential energy of the system of charges. Whereas electric potentail is measured in volt, electric potential energy is measured in joule. You are given a square of each side 1.0 metre with four charges `+1xx10^(-8) C, -2xx10^(-8)C, +3xx10^(-8)C` and `+2xx10^(-8) C` placed at the four corners of the square. With the help of the passage given above, choose the most approprite alternative for each of the following questions :
Electric potentail and electric potential energy

Options

A. both are scalars
B. both are vectors
C. electric potential is scalar and electric potential energy is vector,
D. electric potentail is vector and electric potential energy is scalar.

Explanation

To solve the problem, we need to analyze the electric potential and electric potential energy of the system of charges placed at the corners of a square. Let's break down the solution step by step.

### Step 1: Understanding Electric Potential
The electric potential \( V \) at a point due to a charge \( q \) at a distance \( r \) is given by the formula:
\[ V = \frac{q}{4\pi \epsilon_0 r} \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Analyzing the Given Charges
We have four charges located at the corners of a square with side length \( 1.0 \, \text{m} \):
- Charge \( q_1 = +1 \times 10^{-8} \, \text{C} \)
- Charge \( q_2 = -2 \times 10^{-8} \, \text{C} \)
- Charge \( q_3 = +3 \times 10^{-8} \, \text{C} \)
- Charge \( q_4 = +2 \times 10^{-8} \, \text{C} \)

### Step 3: Calculating the Electric Potential at Each Corner
To find the total electric potential at any corner due to all other charges, we need to calculate the potential contribution from each charge at that corner.

1. **Potential at Corner A (due to charges at B, C, and D):**
- Distance from A to B = 1 m
- Distance from A to C = 1 m (diagonal)
- Distance from A to D = 1 m

The total potential \( V_A \) at corner A is:
\[ V_A = V_{B} + V_{C} + V_{D} \]
where \( V_B, V_C, V_D \) are the potentials due to charges at corners B, C, and D respectively.

2. **Calculating Each Potential:**
- \( V_B = \frac{-2 \times 10^{-8}}{4\pi \epsilon_0 \cdot 1} \)
- \( V_C = \frac{3 \times 10^{-8}}{4\pi \epsilon_0 \cdot 1} \)
- \( V_D = \frac{2 \times 10^{-8}}{4\pi \epsilon_0 \cdot 1} \)

Substitute these values into the equation for \( V_A \).

### Step 4: Summing Up the Potentials
After calculating the individual potentials, sum them up to get the total potential at corner A. Repeat this process for the other corners.

### Step 5: Calculating Electric Potential Energy
The electric potential energy \( U \) of the system of charges is given by the formula:
\[
U = k \sum_{i