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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 determine the position at which the electric field is zero at point A, we need to consider the electric fields created by the two charges at the corners of the square.
Given:
Charge on each object = 83.0 µC
Side length of the square = 0.320 m
Step 1: Calculate the magnitude and direction of each electric field.
The electric field (E) created by a point charge (q) at a distance (r) from the charge can be calculated using Coulomb's Law:
E = k * (q / r^2)
where:
k is the electrostatic constant (k ≈ 8.99 × 10^9 N m^2/C^2)
q is the charge
r is the distance from the charge
For each charge at the corners of the square, the distance (r) from the charge to point A can be found using the Pythagorean theorem:
r = sqrt((0.320 m)^2 + (0.320 m)^2) = 0.453 m
Now we can calculate the magnitude of the electric field (E) created by each charge:
E = (8.99 × 10^9 N m^2/C^2) * (83.0 × 10^-6 C) / (0.453 m)^2
E = 3.608 N/C (rounded to three decimal places)
Since the charges at the corners of the square have the same magnitude of charge, their electric fields will have the same magnitude.
Step 2: Determine the direction of the electric field vectors.
The electric field vectors from the two charges at the corners of the square will be directed toward point A. Therefore, the direction of the net electric field at point A will be the vector sum of the two individual electric fields.
Since the two electric fields have the same magnitude and are directed along the same line, their vector sum will be along that line as well. This means the net electric field at point A is zero when the magnitudes of the electric fields are equal but opposite in direction.
Step 3: Place the third charge at the desired position.
To cancel out the electric field at point A, we need to place the third charge at a position where its electric field is equal in magnitude, but opposite in direction, to the electric field created by the other two charges.
Since the electric fields are directed toward point A, we need to place the third charge above and to the left of point A, so that its electric field points downward and to the right.
Given that we have two charges with the same value and sign at diagonally opposite corners of the square, the position of the third charge with the same sign as the other two will be at the diagonally opposite corner of the square (bottom right corner).
So, place the third charge at the corner of the square diagonally opposite from point A (bottom right corner in this case). The distance between this corner and point A can be found using the Pythagorean theorem:
d = sqrt(0.320 m)^2 + (0.320 m)^2) = 0.453 m
Therefore, the third charge should be placed 0.453 m above and to the left of the corner labeled A so that the electric field is zero at point A.