four equal-magnitude (4uc)charges are placed at the four corners of a square that is 20cm on each side.find the electric field intensity at the center of square , if the charge are all positive.
wouldn't it be zero, due to each E direction symettrical?
To find the electric field intensity at the center of the square, we can calculate the electric field intensity due to each individual charge and then add them together.
Step 1: Calculate the electric field intensity due to one charge at the center.
The electric field intensity due to a point charge is given by the formula:
E = (k * q) / r^2
where E is the electric field intensity, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q is the charge, and r is the distance from the charge.
Since all the charges are equal in magnitude, let's assume each charge to be q.
The distance from one of the charges to the center of the square is half the length of the side, which is 10 cm or 0.1 m (20 cm / 2 = 10 cm).
Substituting the values into the formula, we get:
E1 = (9 x 10^9 N m^2/C^2 * q) / (0.1 m)^2
E1 = (9 x 10^9 N m^2/C^2 * q) / 0.01 m^2
E1 = (9 x 10^9 N m^2/C^2) * q / 0.01 m^2
Step 2: Calculate the total electric field intensity by adding the contributions from all four charges.
Since all four charges are equal in magnitude, the total electric field intensity at the center of the square is four times the electric field intensity due to one charge:
E_total = 4 * E1
Substituting the expression for E1, we get:
E_total = 4 * [(9 x 10^9 N m^2/C^2) * q / 0.01 m^2]
Step 3: Substitute the value of the charge.
Substituting the value of the charge, which is 4uC (4 * 10^-6 C), we get:
E_total = 4 * [(9 x 10^9 N m^2/C^2) * (4 * 10^-6 C) / 0.01 m^2]
Step 4: Simplify the equation and calculate the result.
E_total = (4 * 9 * 10^9 N m^2/C^2) * (4 * 10^-6 C / 0.01 m^2)
E_total = (36 * 10^9 N m^2/C^2) * (4 * 10^-6 C / 0.01 m^2)
E_total = 36 * 4 * (10^9 N m^2/C^2) * (10^-6 C / 10^-2 m^2)
E_total = 144 * 10^3 N/C
Therefore, the electric field intensity at the center of the square is 144,000 N/C.
To calculate the electric field intensity at the center of the square, we can use the principle of superposition. This principle 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 magnitude produced by a single charge at the center of the square. The electric field due to a point charge is given by Coulomb's Law:
E = k * q / r^2
Where:
E is the electric field magnitude,
k is Coulomb's constant (approximately 9 × 10^9 N·m^2/C^2),
q is the magnitude of the charge, and
r is the distance between the charge and the point where we want to calculate the electric field.
In this case, the magnitude of each charge is 4 μC, and the distance from each charge to the center of the square is half the diagonal of the square. Since the square has sides of length 20 cm, the diagonal can be found using the Pythagorean theorem:
Diagonal^2 = Side^2 + Side^2
= 20 cm^2 + 20 cm^2
= 800 cm^2
Diagonal = sqrt(800) cm
≈ 28.28 cm
Therefore, the distance from each charge to the center of the square is half of the diagonal:
r = 28.28 cm / 2
= 14.14 cm
Now, we can calculate the electric field magnitude produced by one charge at the center:
E1 = (9 × 10^9 N·m^2/C^2) * (4 μC) / (14.14 cm)^2
Next, we need to find the direction of each electric field. Since the charges are all positive and located at the corners of the square, the electric fields will point radially outward from each charge.
Since there are four charges, we need to calculate the electric field magnitude for each charge and then sum them up vectorially. However, since the charges are at opposite corners of the square, the electric fields due to these charges will have equal magnitude but opposite direction, effectively canceling each other out.
The remaining two charges are adjacent to each other along diagonals of the square. Since the distance from each of these charges to the center is also 14.14 cm, the electric fields due to these charges will also have equal magnitude and the same direction.
Therefore, to find the net electric field at the center of the square, we only need to consider the two adjacent charges. The electric field produced by one charge will be in the x-direction, and the other will be in the y-direction. The net electric field can be calculated using vector addition:
E_net = sqrt(E_x^2 + E_y^2)
Now, let's calculate E_x:
E_x = E1 * cos(45°)
= E1 * (1/√2)
Similarly, let's calculate E_y:
E_y = E1 * sin(45°)
= E1 * (1/√2)
Finally, we can calculate the net electric field:
E_net = sqrt((E1 * (1/√2))^2 + (E1 * (1/√2))^2)
= sqrt(2 * (E1^2))
Substitute the value of E1 and calculate E_net to find the electric field intensity at the center of the square.