Each charge of 3 micro coulomb are placed at 3 corners of a square whose diagonal is 6cm long. Find the field intensity at the point of intersection of diagonals

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To find the field intensity at the point of intersection of diagonals, we will use the principle of superposition. This principle states that the total field intensity at a point due to multiple charges is the vector sum of the individual field intensities produced by each charge.

Given:
Charge of each corner = 3 micro Coulomb (3 μC)
Diagonal of the square = 6 cm

Let's assume the side length of the square is 'a'.

Since the diagonals intersect at the center of the square, the distance (r) from the center to any charge is:

r = a/2

By the Pythagorean theorem, the diagonal length (d) is related to the side length (a):

d^2 = a^2 + a^2
36 = 2a^2
a^2 = 18
a = sqrt(18) = 3√2

Now, we can calculate the field intensity at the point of intersection of diagonals.

The electric field intensity (E) at a distance (r) from a point charge (Q) is given by:

E = k * (Q / r^2)

Where k is the electrostatic constant, equal to 9 x 10^9 Nm^2/C^2.

Let's find the field intensity at the center due to each charge:

E1 = (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((a/2)^2))
E2 = (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((a/2)^2))
E3 = (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((a/2)^2))
E4 = (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((a/2)^2))

Since the charges are located on the corners of the square, the total field intensity at the center is the vector sum of the individual field intensities:

E_total = E1 + E2 + E3 + E4

Now let's substitute the values and calculate:

E_total = E1 + E2 + E3 + E4
E_total = 4 * (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((a/2)^2))
E_total = 36 * (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((3√2/2)^2))

Simplifying,

E_total = 36 * (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / ((3/2)^2))
E_total = 36 * (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) / (9/4))
E_total = 36 * (9 x 10^9 Nm^2/C^2) * ((3 x 10^(-6) C) * (4/9))
E_total = 36 * (4 x 10^9 Nm^2/C^2) * (3 x 10^(-6) C)

Now, let's calculate the final result:

E_total = 36 * 4 * 3 x 10^3 N/C
E_total = 43200 N/C

Therefore, the field intensity at the point of intersection of diagonals is 43200 N/C.

To find the electric field intensity at the point of intersection of the diagonals, we can use the principle of superposition. Electric field is a vector quantity, so we need to consider the contributions from each charge separately and then add them together.

Let's start by labeling the charges and the length of the diagonal for reference:

A --- x --- B
| |
y P z
| |
C --- w --- D

We have four charges placed at the corners of the square: A, B, C, and D. The point of intersection of the diagonals is denoted by P.

Given that each charge has a magnitude of 3 microcoulombs (3 µC) and the diagonal length is 6 cm, we can divide each diagonal into segments with lengths of x, y, z, and w according to the diagram above.

Now, let's calculate the electric field contribution from each charge and then add them up:

1. Charge at A:
The electric field at P due to A can be calculated using Coulomb's Law:
E_A = k * (q_A / r_A^2)
where k is the electrostatic constant (k = 9 × 10^9 N m²/C²), q_A is the charge at A (3 µC), and r_A is the distance between A and P.

Since A and P are opposite corners of the square's diagonal, we can use Pythagoras' theorem to find r_A:
r_A = √(x^2 + y^2)

2. Charge at B:
Similarly, the electric field at P due to B can be calculated as:
E_B = k * (q_B / r_B^2)
where q_B is the charge at B (3 µC) and r_B is the distance between B and P.
Similarly to A, we can use Pythagoras' theorem to find r_B:
r_B = √(z^2 + y^2)

3. Charge at C:
The electric field at P due to C can be calculated as:
E_C = k * (q_C / r_C^2)
where q_C is the charge at C (3 µC) and r_C is the distance between C and P.
Again, using Pythagoras' theorem to find r_C:
r_C = √(x^2 + w^2)

4. Charge at D:
The electric field at P due to D can be calculated as:
E_D = k * (q_D / r_D^2)
where q_D is the charge at D (3 µC) and r_D is the distance between D and P.
Applying Pythagoras' theorem to find r_D:
r_D = √(z^2 + w^2)

Finally, to find the net electric field at P, we need to sum up the contributions from each charge:
E_total = E_A + E_B + E_C + E_D

By substituting the appropriate values into the equations and performing the calculations, we can find the field intensity at the point of intersection of diagonals.