3 equal charges (all +Q) are located at A, B and C. ABC is an equilateral triangle with sides d. Point A is located at (0,0), point B at (d,0), point C at (d/2,d3√/2). Point P is inside the triangle such that AP=BP=CP. What is the electric field at P - we want the magnitude and the direction.
24Q4πϵ0d2 yˆ
0
24Q4πϵ0d2 (-yˆ)
12Q4πϵ0d2 yˆ
To find the electric field at point P, we need to consider the contribution from each of the charges located at A, B, and C. The electric field at P due to a single charge can be calculated using Coulomb's Law.
1. Electric Field due to charge at A:
The position vector for point A is (0,0). The position vector for point P is (x,y). The distance between A and P is given by r₁ = sqrt(x² + y²). Since AP = BP = CP, the distance from P to any of the charges is the same.
The electric field at P due to the charge at A is given by the formula:
E₁ = (k * Q / r₁²) * r̂₁
Here, k is the Coulomb constant, Q is the magnitude of the charge at A, r̂₁ is the unit vector in the direction of the position vector from A to P.
2. Electric Field due to charge at B:
The position vector for point B is (d, 0). The position vector for point P is still (x, y). The distance between B and P is given by r₂ = sqrt((x - d)² + y²).
The electric field at P due to the charge at B is given by the formula:
E₂ = (k * Q / r₂²) * r̂₂
Here, r̂₂ is the unit vector in the direction of the position vector from B to P.
3. Electric Field due to charge at C:
The position vector for point C is (d/2, d√3/2). The position vector for point P is still (x, y). The distance between C and P is given by r₃ = sqrt((x - d/2)² + (y - d√3/2)²).
The electric field at P due to the charge at C is given by the formula:
E₃ = (k * Q / r₃²) * r̂₃
Here, r̂₃ is the unit vector in the direction of the position vector from C to P.
To get the total electric field at P, we need to sum up the contributions from the charges at A, B, and C. Since these charges are equal in magnitude (+Q), we can add the electric fields as vectors.
E_total = E₁ + E₂ + E₃
Finally, we can calculate the magnitude and direction of the total electric field at P by using the formulas:
Magnitude: |E_total| = sqrt((Ex)^2 + (Ey)^2)
Direction: E_total / |E_total|
To find the electric field at point P, we can use the superposition principle to calculate the contribution of each of the charges at points A, B, and C, and then add them together.
1. Electric field due to charge at point A:
The distance from point A to P is AP, which we know is equal to BP and CP. Let's denote this distance as r. The electric field due to a single charge at point A can be calculated using Coulomb's law:
E_A = k * Q / r^2, where k is the electrostatic constant, Q is the charge, and r is the distance.
2. Electric field due to charge at point B:
The distance from point B to P is BP, which is also equal to AP and CP. Let's denote this distance as r as well. The electric field due to a single charge at point B can be calculated in the same way:
E_B = k * Q / r^2.
3. Electric field due to charge at point C:
The distance from point C to P is CP, which is again equal to AP and BP. Let's denote this distance as r too. The electric field due to a single charge at point C can be calculated as follows:
E_C = k * Q / r^2.
Now, since AP = BP = CP, the electric field vectors at P due to charges A, B, and C will have components in the same direction. Therefore, we can simply add the magnitudes of these electric field vectors to find the total electric field at P:
E_total = E_A + E_B + E_C.
Since the charges at A, B, and C are all +Q, their electric fields will be directed towards P along the negative y-axis, i.e., in the downward direction. Therefore, the direction of the total electric field vector at P will be downward.
The magnitude of the total electric field at P will be:
E_total = 3 * (k * Q / r^2) = 3kQ / r^2.
Substituting k = 1 / (4πɛ₀), where ɛ₀ is the permittivity of free space, we get:
E_total = 3Q / (4πɛ₀ * r^2).
Thus, the magnitude of the electric field at P is 3Q / (4πɛ₀ * r^2), directed downward along the negative y-axis.
Therefore, the correct answer is:
24Q / (4πɛ₀ * d^2) in the -ŷ direction or -ŷ 24Q / (4πɛ₀ * d^2).