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Two tiny objects with equal charges of 83.0 µC are placed at two corners of a square with sides of 0.320 m, as shown. How far above and to the left of the corner of the square labeled A would you place a third small object with the same charge so that the electric field is zero at A?
To find the position where the electric field is zero at point A, we need to calculate the distance and direction from the corner of the square labeled A. Here's how you can do it step by step:
Step 1: Understand the problem
In this problem, we have two identical charges placed at two corners of a square, and we need to find the position of a third charge where the electric field is zero at the corner labeled A.
Step 2: Identify the given information
We are given that the charges at the other corners have a magnitude of 83.0 µC and that the sides of the square have a length of 0.320 m.
Step 3: Determine the direction of the electric field
Since we want the electric field to be zero at point A, we need to place the third charge in such a way that the electric fields from the other two charges cancel each other out. To do this, the direction of the electric field from the two charges must be opposite to each other at point A.
Step 4: Calculate the distance from the corner of the square labeled A
To determine the distance from the corner of the square labeled A, we can use the Pythagorean theorem. Since the square has equal sides, the diagonal connecting the corner labeled A to the other corner is equal to the side length multiplied by the square root of 2.
Diagonal length = Side length * sqrt(2)
Diagonal length = 0.320 m * sqrt(2)
Step 5: Calculate the position of the third charge
To find the position of the third charge, we need to identify the direction and distance from the corner of the square labeled A. From the previous step, we know the diagonal length connecting the corner of the square labeled A to the other corner. To make the electric field zero, we need to place the third charge in such a way that it forms a triangle with the other two charges, with the diagonal of the square as its base.
Using trigonometry, the x-coordinate of the third charge can be calculated as follows:
x = (Diagonal length / 2) * cos(45°)
x = (0.320 m * sqrt(2) / 2) * cos(45°)
Similarly, the y-coordinate of the third charge can be calculated as:
y = (Diagonal length / 2) * sin(45°)
y = (0.320 m * sqrt(2) / 2) * sin(45°)
Step 6: Calculate the final answer
Now, you can plug in the values and calculate the x and y coordinates separately:
x = (0.320 m * sqrt(2) / 2) * cos(45°)
y = (0.320 m * sqrt(2) / 2) * sin(45°)
Evaluate the cos(45°) and sin(45°) which are equal to 0.7071:
x = (0.320 m * sqrt(2) / 2) * 0.7071
y = (0.320 m * sqrt(2) / 2) * 0.7071
Now, calculate the values of x and y:
x = (0.320 m * 1.414) / 2
y = (0.320 m * 1.414) / 2
x ≈ 0.226 m
y ≈ 0.226 m
So, the third charge should be placed approximately 0.226 m above and 0.226 m to the left of the corner labeled A to make the electric field zero at point A.