Identical point charges of +2.9 µC are fixed to three of the four corners of a square. What is the magnitude of the negative point charge that must be fixed to the fourth corner, so that the charge at the diagonally opposite corner experiences a net force of zero?
To find the magnitude of the negative point charge at the fourth corner, we can use Coulomb's Law. Coulomb's Law relates the magnitude of the electrostatic force between two point charges to the charges and the distance between them.
Let's break down the steps to solve this problem:
Step 1: Understand the problem.
We have three identical point charges of +2.9 µC fixed to three corners of a square. We need to find the magnitude of the negative point charge at the fourth corner such that the charge at the diagonally opposite corner experiences a net force of zero.
Step 2: Recall Coulomb's Law.
Coulomb's Law states that the electrostatic force (F) between two point charges is directly proportional to the product of their charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. Mathematically, it can be expressed as:
F = (k * |q1 * q2|) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (k ≈ 8.99 x 10^9 N.m^2/C^2)
q1 and q2 are the magnitudes of the charges
r is the distance between the charges
Step 3: Analyze the problem.
Since the charge at the diagonally opposite corner experiences a net force of zero, the forces due to the other three charges must balance out.
Step 4: Set up the equation.
Let's consider the charge at the diagonal corner as q4 and the charges at the other three corners as q1, q2, and q3. The distance between opposite corners is the same for all four side lengths of the square. Mathematically, we can set up the equation as:
F1 + F2 + F3 = 0
Using Coulomb's Law, we can represent these forces as:
(k * |q1 * q4|) / r^2 + (k * |q2 * q4|) / r^2 + (k * |q3 * q4|) / r^2 = 0
Step 5: Solve for q4.
Rearranging the equation, we get:
(k/r^2) * (|q1 * q4| + |q2 * q4| + |q3 * q4|) = 0
Since k, r, and |q1 * q4|, |q2 * q4|, and |q3 * q4| are positive, we can solve for q4 by simply setting the sum of the charges at the other corners equal to zero:
|q1 * q4| + |q2 * q4| + |q3 * q4| = 0
Step 6: Substitute the values.
Substituting the given charge magnitude (+2.9 µC) for q1, q2, and q3, the equation becomes:
|(+2.9 µC * q4)| + |(+2.9 µC * q4)| + |(+2.9 µC * q4)| = 0
Simplifying further:
|2.9 µC * q4| + |2.9 µC * q4| + |2.9 µC * q4| = 0
Step 7: Solve for q4.
By combining the terms, we have:
3 * |2.9 µC * q4| = 0
Since the magnitude of any charge cannot be negative, the only solution to this equation is when the magnitude of each charge is zero. Therefore, the magnitude of the negative point charge at the fourth corner that will balance out the other charges is 0 µC.
In conclusion, the magnitude of the negative point charge that must be fixed to the fourth corner, so that the charge at the diagonally opposite corner experiences a net force of zero, is 0 µC.