Four charges with equal magnitudes of 6.50 x 10-15 C are placed at the corners of a rectangle. The lengths of the sides of the rectangles are 4.07 cm and 5.28 cm. Find the magnitude of the electric field at the center of the rectangle, assuming that three charges have the same sign and the remaining one has the opposite sign.

Question

Four charges with equal magnitudes of 6.50 x 10-15 C are placed at the corners of a rectangle. The lengths of the sides of the rectangles are 4.07 cm and 5.28 cm. Find the magnitude of the electric field at the center of the rectangle, assuming that three charges have the same sign and the remaining one has the opposite sign.

Answer 1

To find the magnitude of the electric field at the center of the rectangle, we can sum up the electric fields due to each charge and find the resultant. The formula for the electric field due to a point charge is given by:

E = k * (Q / r^2)

Where:
- E is the electric field
- k is the Coulomb's constant, approximately equal to 9 x 10^9 Nm^2/C^2
- Q is the charge
- r is the distance from the charge to the point where we want to find the electric field

Let's calculate the electric field due to each charge and then sum them up:

1. Charge at the top left corner:
The distance from the charge to the center of the rectangle is the diagonal length of the rectangle. We can calculate it using the Pythagorean theorem:
d1 = √(4.07 cm)^2 + (5.28 cm)^2 = 6.67 cm = 0.0667 m

Using the formula for the electric field, we have:
E1 = k * (6.50 x 10^-15 C / (0.0667 m)^2)

2. Charge at the top right corner:
The distance from the charge to the center of the rectangle is the same as the previous one:
d2 = 0.0667 m

Using the formula for the electric field, we have:
E2 = k * (6.50 x 10^-15 C / (0.0667 m)^2)

3. Charge at the bottom right corner:
The distance from the charge to the center of the rectangle can be calculated similarly:
d3 = 0.0667 m

Using the formula for the electric field, we have:
E3 = k * (6.50 x 10^-15 C / (0.0667 m)^2)

4. Charge at the bottom left corner:
The distance from the charge to the center of the rectangle can be calculated similarly:
d4 = 0.0667 m

Using the formula for the electric field, we have:
E4 = k * (-6.50 x 10^-15 C / (0.0667 m)^2) (since this charge has the opposite sign)

Now, we can sum up the electric fields:
E_total = E1 + E2 + E3 + E4

After calculating this sum, we will find the magnitude of the electric field at the center of the rectangle.