Four charges with equal magnitudes of 7.81 × 10-12 C are placed at the corners of a rectangle. The lengths of the sides of the rectangles are 2.52 cm and 4.02 cm. Find the magnitude of the electric field at the center of the rectangle in Figures a and b. img166.imageshack.us/img166/3623/1821figvn8.gif

To find the magnitude of the electric field at the center of the rectangle, we need to calculate the electric field contributed by each charge individually and then add them up.

First, let's calculate the electric field due to a single charge at the center of the rectangle. The electric field due to a point charge can be calculated using the formula:

E = k * (q / r^2)

where E is the electric field, k is the electrostatic constant (9 × 10^9 N.m^2/C^2), q is the charge, and r is the distance from the charge to the point where we want to calculate the electric field.

In this case, the distance from each charge to the center of the rectangle is the same, which is the diagonal of the rectangle. We can find this distance using the Pythagorean theorem:

d = sqrt((l/2)^2 + (w/2)^2)

where l is the length of the rectangle and w is the width of the rectangle.

Let's calculate the value of d:

l = 4.02 cm = 0.0402 m
w = 2.52 cm = 0.0252 m

d = sqrt((0.0402/2)^2 + (0.0252/2)^2)
= sqrt(0.01020101 + 0.00317552)
= sqrt(0.01337653)
≈ 0.1157 m

Now, we can calculate the electric field due to a single charge at the center of the rectangle:

E = (9 × 10^9 N.m^2/C^2) * (7.81 × 10^-12 C) / (0.1157 m)^2
= (9 × 10^9 N.m^2/C^2) * (7.81 × 10^-12 C) / 0.01337249 m^2
= 5.2144 × 10^6 N/C

Therefore, the magnitude of the electric field due to a single charge at the center of the rectangle is 5.2144 × 10^6 N/C.

Since there are four charges with the same magnitude, the total electric field at the center of the rectangle is four times the electric field due to a single charge:

E_total = 4 * (5.2144 × 10^6 N/C)
= 20.8576 × 10^6 N/C
= 2.08576 × 10^7 N/C

Therefore, the magnitude of the electric field at the center of the rectangle in Figures a and b is 2.08576 × 10^7 N/C.