To solve this problem, we need to analyze the forces acting on the point charge as it moves towards the center of the concentric rings. The setup involves two rings: one with a positive charge and the other with a negative charge. The goal is to determine the minimum velocity required for a point charge to reach the center of these rings, starting from a distance of 3r along the axis. Let's break this down step by step.
Understanding the Electric Field
First, we need to calculate the electric field produced by both rings at a point along the axis. The electric field due to a charged ring at a distance z from its center along its axis is given by the formula:
E = (k * Q * z) / (z^2 + R^2)^(3/2)
where:
- k is Coulomb's constant (approximately 8.99 x 10^9 N m²/C²),
- Q is the total charge of the ring,
- R is the radius of the ring, and
- z is the distance from the center of the ring to the point where the electric field is being calculated.
Calculating the Electric Fields
For the inner ring (radius r, charge q), at a distance z = 3r:
E1 = (k * q * 3r) / ((3r)^2 + r^2)^(3/2)
This simplifies to:
E1 = (k * q * 3r) / (10r^2)^(3/2) = (k * q * 3r) / (1000 * r^3) = (3kq) / (1000r^2)
For the outer ring (radius 4r, charge -8q), at the same distance z = 3r:
E2 = (k * (-8q) * 3r) / ((3r)^2 + (4r)^2)^(3/2)
This simplifies to:
E2 = (k * (-8q) * 3r) / (25r^2)^(3/2) = (k * (-8q) * 3r) / (125 * r^3) = (-24kq) / (125r^2)
Net Electric Field
The total electric field at the point 3r from the center is the sum of the electric fields from both rings:
E_total = E1 + E2 = (3kq) / (1000r^2) + (-24kq) / (125r^2)
Finding a common denominator (1000r^2) gives:
E_total = (3kq - 192kq) / (1000r^2) = (-189kq) / (1000r^2)
Force on the Point Charge
The force acting on the point charge of mass m and charge -q is given by:
F = qE_total = -q * (-189kq) / (1000r^2) = (189kq^2) / (1000r^2)
Energy Considerations
To find the minimum velocity, we can use the concept of energy conservation. The point charge starts with kinetic energy and gains potential energy as it moves against the electric field:
KE_initial + PE_initial = KE_final + PE_final
At the starting point (3r), the potential energy is zero (we can set it as a reference), and at the center (0), it will be:
PE_final = -q * V_center
Where V_center is the potential at the center due to both rings. The potential due to a ring is given by:
V = k * Q / R
Calculating the potentials at the center from both rings and summing them gives:
V_center = k * (q/r + (-8q)/(4r)) = k * (q/r - 2q/r) = -kq/r
Thus, the potential energy at the center is:
PE_final = -q * (-kq/r) = kq^2/r
Setting Up the Equation
Now, we can set up the energy conservation equation:
1/2 mv^2 = kq^2/r
Solving for v gives:
v = sqrt((2kq^2)/(mr))
Final Result
The minimum velocity with which the point charge of mass m and charge -q should be projected from a distance of 3r from the center of the rings along the axis to reach the center is:
v = sqrt((2kq^2)/(mr))
This result shows how the interplay of electric forces and energy conservation principles allows us to determine the necessary conditions for the charge to reach the center of the rings. If you have any further questions or need clarification on any part of this explanation, feel free to ask!
