Askiitians Tutor Team
Last Activity: 5 Months ago
To solve the problem of keeping particles A and B at rest under the influence of electrical forces, we need to consider the forces acting on them due to their charges and the charge of particle C. Let's break this down step by step.
Understanding the Forces
Particles A and B have charges q and 2q, respectively, and they are separated by a distance d. The force between two charged particles can be calculated using Coulomb's Law, which states:
F = k * |q1 * q2| / r²
where:
- F is the force between the charges,
- k is Coulomb's constant,
- q1 and q2 are the magnitudes of the charges, and
- r is the distance between the charges.
Setting Up the Problem
For particles A and B to remain at rest, the net force acting on each particle must be zero. This means that the force exerted by particle C on A must balance the electrostatic repulsion between A and B, and similarly for particle B.
Calculating Forces
Let’s denote the charge of particle C as Q and its position on the table as x, where x is the distance from particle A. The distance from particle B to particle C will then be (d - x).
The force exerted by particle C on particle A is given by:
F_CA = k * |q * Q| / x²
The force exerted by particle C on particle B is:
F_CB = k * |2q * Q| / (d - x)²
Establishing Equilibrium
For particle A to be in equilibrium, the force from C must balance the repulsive force between A and B:
F_CA = F_AB
Where:
F_AB = k * |q * 2q| / d² = 2k * q² / d²
Thus, we have:
k * |q * Q| / x² = 2k * q² / d²
We can simplify this equation by canceling out k and q (assuming q is not zero):
|Q| / x² = 2q / d²
From this, we can express Q in terms of x:
Q = 2q * x² / d²
For Particle B
Similarly, for particle B to be in equilibrium, we set up the equation:
F_CB = F_AB
Which gives us:
k * |2q * Q| / (d - x)² = 2k * q² / d²
Again, simplifying this leads to:
|Q| / (d - x)² = q / d²
From this, we can express Q in terms of (d - x):
Q = q * (d - x)² / d²
Finding the Charge and Position
Now we have two expressions for Q:
Q = 2q * x² / d²
Q = q * (d - x)² / d²
Setting these equal to each other gives:
2q * x² = q * (d - x)²
Dividing both sides by q (assuming q is not zero) leads to:
2x² = (d - x)²
Expanding the right side:
2x² = d² - 2dx + x²
Rearranging gives:
x² + 2dx - d² = 0
This is a quadratic equation in x. We can solve for x using the quadratic formula:
x = [-2d ± √(4d² + 4d²)] / 2
Which simplifies to:
x = -d ± √(2)d
Since x must be positive, we take:
x = -d + √(2)d = (√2 - 1)d
Final Charge Calculation
Now substituting x back into the expression for Q:
Q = 2q * [(√2 - 1)d]² / d²
Calculating this gives:
Q = 2q * (2 - 2√2 + 1) = 2q(3 - 2√2)
Thus, the charge of particle C should be:
Q = 2q(3 - 2√2)
Position of Particle C
Finally, particle C should be clamped at a distance of:
x = (√2 - 1)d
In summary, to keep particles A and B at rest, particle C should have a charge of 2q(3 - 2√2) and should be positioned at a distance of (√2 - 1)d from particle A on the table.
