Two point charges Q1 = -26.8 micro-Coulombs and Q2 = +38.5 micro-Coulombs are located on a vertical line with Q1 below Q2. The coordinates of Q1 is (0.0, -6.8 cm). The two charges are 28.6 cm apart. Determine the coordinates of the location where the net electric field due to Q1 and Q2 is zero.
To find the coordinates of the location where the net electric field due to Q1 and Q2 is zero, we can use the principle of the superposition of electric fields. The net electric field at any point in space is the vector sum of the individual electric fields created by each charge. At the point where the net electric field is zero, the magnitudes of the individual electric fields are equal but opposite in direction.
Let's assume that the coordinate of the unknown point is (x, y). We need to calculate both the x-coordinate and the y-coordinate.
First, let's calculate the electric field due to charge Q1 at the unknown point. The electric field due to a point charge is given by the formula:
E1 = k * |Q1| / r1^2
where k is the electrostatic constant (k ≈ 8.99 x 10^9 N·m^2/C^2), |Q1| is the magnitude of charge Q1, and r1 is the distance from Q1 to the unknown point.
Given that Q1 = -26.8 μC and the distance from Q1 to the unknown point is y + 6.8 cm, we can calculate the electric field E1 due to Q1 at the unknown point.
E1 = (8.99 x 10^9 N·m^2/C^2) * (26.8 x 10^-6 C) / (y + 6.8 x 10^-2 m)^2
Similarly, we can calculate the electric field due to charge Q2 at the unknown point. The electric field due to a point charge is given by the same formula:
E2 = k * |Q2| / r2^2
where |Q2| is the magnitude of charge Q2 and r2 is the distance from Q2 to the unknown point.
Given that Q2 = +38.5 μC and the distance from Q2 to the unknown point is 28.6 cm - y, we can calculate the electric field E2 due to Q2 at the unknown point.
E2 = (8.99 x 10^9 N·m^2/C^2) * (38.5 x 10^-6 C) / (28.6 x 10^-2 m - y)^2
Since we want the net electric field to be zero, the magnitudes of E1 and E2 must be equal. Therefore, we can set E1 equal to E2:
(8.99 x 10^9 N·m^2/C^2) * (26.8 x 10^-6 C) / (y + 6.8 x 10^-2 m)^2 = (8.99 x 10^9 N·m^2/C^2) * (38.5 x 10^-6 C) / (28.6 x 10^-2 m - y)^2
Now, we can solve this equation to find the value of y. Once we have the value of y, we can substitute it back into either equation to find the corresponding x-coordinate.
After finding the values of x and y, we will have the coordinates of the location where the net electric field due to Q1 and Q2 is zero.
To determine the location where the net electric field due to Q1 and Q2 is zero, 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 individual electric fields due to each charge.
Let's assume the location where the net electric field is zero has coordinates (x, y). We can find the electric field at that point due to each charge and then add them together.
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,
k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2),
Q is the charge,
and r is the distance from the charge to the point.
First, let's calculate the electric field due to Q1:
Q1 = -26.8 μC
Coordinates of Q1 = (0.0, -6.8 cm)
Distance between Q1 and the point = y + 6.8 cm
E1 = k * (Q1 / r^2),
E1 = k * (Q1 / (y + 6.8 cm)^2)
Next, let's calculate the electric field due to Q2:
Q2 = +38.5 μC
Coordinates of Q2 = (x, y)
Distance between Q2 and the point = x
E2 = k * (Q2 / r^2),
E2 = k * (Q2 / x^2)
Since the net electric field is zero, the sum of the electric fields due to Q1 and Q2 should be zero:
E1 + E2 = 0
Now, we can substitute the expressions for E1 and E2:
k * (Q1 / (y + 6.8 cm)^2) + k * (Q2 / x^2) = 0
Solving this equation will give us the values of x and y, which represent the coordinates of the location where the net electric field due to Q1 and Q2 is zero.