Charges are placed on an equilateral triangle (all angles are equal and equal to 60 degrees). The length of each side of the triangle is 5.7 meters. A charge of +7.1 micro-Coulombs is placed at the bottom corner of the triangle and another charge of +7.1 micro-Coulombs is placed in the other corner at the bottom of the triangle. Another charge is placed halfway between the two other charges (directly below the other corner of the triangle - see sketch). What does this charge need to be such that the total electric field in the third, unoccupied corner, of the triangle is equal to zero? Answer in micro-Coulombs.
To find the charge needed in the third, unoccupied corner of the triangle such that the total electric field in that corner is equal to zero, we can use the principle of superposition. This principle states that the electric field at a point due to multiple charges is the vector sum of the electric fields produced by each individual charge.
In this case, we have two charges of +7.1 micro-Coulombs each placed at the bottom corners of the equilateral triangle. Let's label these charges as q1 and q2.
The electric field produced by a point charge can be calculated using Coulomb's Law. The formula to find the electric field (E1) produced by a charge (q1) at a point is given by:
E1 = k * q1 / r1²
where k is the electrostatic constant (k ≈ 9 × 10^9 Nm²/C²), q1 is the charge, and r1 is the distance between the charge and the point where we want to find the electric field.
Now, let's consider the unoccupied corner of the triangle as point P and find the electric field (E1) produced by charge q1 at point P. The distance between q1 and P is 5.7 meters (since each side of the equilateral triangle is 5.7 meters). Therefore, we have:
E1 = k * q1 / (5.7)^²
Similarly, the electric field (E2) produced by charge q2 at point P is also calculated using the same formula, but now the distance (r2) is also 5.7 meters.
E2 = k * q2 / (5.7)^²
Since we want the total electric field at point P to be zero, we can write the equation:
E1 + E2 = 0
Substituting the previously calculated values, we have:
k * q1 / (5.7)^² + k * q2 / (5.7)^² = 0
We know that q1 and q2 are both +7.1 micro-Coulombs. Substituting those values and simplifying the equation:
k * (7.1 * 10^-6) / (5.7)^² + k * (7.1 * 10^-6) / (5.7)^² = 0
Dividing both sides of the equation by k / (5.7)^²:
2 * (7.1 * 10^-6) / (5.7)^² = 0
Multiply both sides by (5.7)^² / 2:
q = - (7.1 * 10^-6) * (5.7)^² / 2
Calculating the value, we find:
q ≈ -1.33 micro-Coulombs
Therefore, the charge needed in the third, unoccupied corner of the triangle to make the total electric field in that corner equal to zero is approximately -1.33 micro-Coulombs.