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
Well, let me tell you, the electric field at the center of the rectangle is shocking! Get it? Electric shock? Anyway, to find the magnitude of the electric field, we can use the formula:
E = k * (q1/r1^2 + q2/r2^2 + q3/r3^2 + q4/r4^2)
Where E is the electric field, k is the Coulomb's constant (9 × 10^9 Nm^2/C^2), q1, q2, q3, and q4 are the charges, and r1, r2, r3, and r4 are the distances from each charge to the center.
Now, let's calculate the distances first. The distances from the charges to the center of the rectangle are:
r1 = r3 = 2.02 cm
r2 = r4 = 1.26 cm
Substituting the values into the formula, we get:
E = (9 × 10^9 Nm^2/C^2) * [(7.81 × 10^-12 C)/(2.02 cm)^2 + (7.81 × 10^-12 C)/(1.26 cm)^2 + (7.81 × 10^-12 C)/(2.02 cm)^2 + (7.81 × 10^-12 C)/(1.26 cm)^2]
Now, let me calculate it for you! *calculating sounds*
Drumroll, please!
The magnitude of the electric field at the center of the rectangle is approximately 13.4 N/C.
Remember, this is just a rough answer, so don't get too charged up about it!
To find the magnitude of the electric field at the center of the rectangle, we can use the principle of superposition, which states that the total electric field at a point due to multiple charges is the vector sum of the electric fields produced by each individual charge.
First, let's calculate the electric field produced by a single charge at the center of the rectangle. The electric field due to a point charge at a distance r is given by Coulomb's law:
E = k * q / r^2
where E is the electric field, k is the Coulomb's constant (9 × 10^9 N·m^2/C^2), q is the charge, and r is the distance from the charge.
In this case, all the charges have the same value, so we can use the formula to find the electric field produced by one charge (E1) and then multiply it by the number of charges (4).
Step 1: Calculate the distance from the center of the rectangle to each corner. Since it is a rectangle, the diagonal passing through the center is equal to the diagonal of the rectangle itself.
Using the Pythagorean theorem, we can find the diagonal distance (d) between the two corners:
d = √(2.52^2 + 4.02^2)
Step 2: Calculate the electric field (E1) produced by one charge at the center of the rectangle.
E1 = k * q / d^2
Step 3: Multiply E1 by the number of charges (4) to get the total electric field at the center of the rectangle.
E_total = 4 * E1
By using the given values and performing the calculations, we can find the magnitude of the electric field at the center of the rectangle.