Four point charges have the same magnitude of `2.4 xx 10^(-12) C` and are fixed to the corners of a square that is 4.0 cm on a side. Three of the charges are positive and one is negative. Determine the magnitude of the net electric field that exists at the center of the square.

Options

A. 14N/C
B. 27 N/C
C. 42 N/C
D. 54 N/C

Answer

To determine the magnitude of the net electric field at the center of the square formed by four point charges, we will follow these steps:

### Step 1: Identify the Charges and Their Positions
We have four point charges:
- Three positive charges (+2.4 x 10^(-12) C)
- One negative charge (-2.4 x 10^(-12) C)

These charges are located at the corners of a square with a side length of 4.0 cm.

### Step 2: Calculate the Distance from the Center to Each Charge
The distance from the center of the square to each corner (where the charges are located) can be calculated using the Pythagorean theorem. The distance \( r \) from the center to a corner is given by:

\[ r = \frac{\sqrt{(4.0 \, \text{cm})^2 + (4.0 \, \text{cm})^2}}{2} = \frac{\sqrt{32}}{2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \, \text{cm} = 0.02\sqrt{2} \, \text{m} \approx 0.0283 \, \text{m} \]

### Step 3: Calculate the Electric Field Due to Each Charge
The electric field \( E \) due to a point charge is given by the formula:

\[ E = \frac{k |q|}{r^2} \]

Where:
- \( k = 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \) (Coulomb's constant)
- \( |q| = 2.4 \times 10^{-12} \, \text{C} \)
- \( r = 0.0283 \, \text{m} \)

Calculating the electric field due to one positive charge:

\[ E_{+} = \frac{(8.99 \times 10^9) \times (2.4 \times 10^{-12})}{(0.0283)^2} \]

Calculating \( (0.0283)^2 \):

\[ (0.0283)^2 \approx 0.00080089 \, \text{m}^2 \]

Now substituting back into the electric field equation:

\[ E_{+} = \frac{(8.99 \times 10^9) \times (2.4 \times 10^{-12})}{0.00080089} \approx \frac{2.1576 \times 10^{-2}}{0.00080089} \approx 26.94 \, \text{N/C} \]

### Step 4: Calculate the Electric Field Due to the Negative Charge
The electric field due to the negative charge will have the same magnitude but will point towards the negative charge:

\[ E_{-} = \frac{(8.99 \times 10^9) \times (2.4 \times 10^{-12})}{(0.0283)^2} \approx 26.94 \, \text{N/C} \]

### Step 5: Determine the Net Electric Field at the Center
Due to symmetry, the electric fields from the three positive charges will add up, while the electric field from the negative charge will point towards it.

The net electric field \( E_{net} \) can be calculated as follows:

1. The electric fields from the three positive charges will combine vectorially. Since they are symmetrically placed, their contributions in the direction towards the center will add up.
2. The negative charge will contribute an electric field pointing towards it.

Since the positive charges are symmetrically placed, the resultant electric field from the three positive charges will be:

\[ E_{net} = 3E_{+} - E_{-} \]

Substituting the values:

\[ E_{net} = 3(26.94) - 26.94 = 53.88 - 26.94 = 26.94 \, \text{N/C} \]

### Final Result
The magnitude of the net electric field at the center of the square is approximately:

\[ E_{net} \approx 54 \, \text{N/C} \]