A negatively charged metal ball of negligible mass is separated from a negatively charged
rectangular block (assume the charge cannot escape) by an initial linear distance ‘d’. On the other
side, the metal ball sits at the start of a semicircular loop of radius’R’, made of wood.
Task a. Find the field due to the charged rectangular block and at what distance from the ball the
ball enters the field of the block to experience a force to just move the same (neglect friction).
Task b. As the block moves towards the ball with a constant velocity ‘v’,find an expression for
the velocity which will enable the ball to rise to the top of the loop.
Task c. As the block moves towards the ball with a constant acceleration ‘a’, find an expression
for the ‘a’ which will enable the ball to rise to the top of the loop.
Task d. calculate the charge and dimensions of the block for which the ball will be driven to the
centre of the semicircle without moving the block.
Task a:
To find the field due to the charged rectangular block at a distance from the metal ball, you can use Coulomb's law. Coulomb's law states that the magnitude of the electric field produced by a point charge is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge.
1. Calculate the electric field at the position of the ball using Coulomb's law:
- Find the magnitude of the charge on the rectangular block.
- Calculate the distance between the block and the ball.
- Use Coulomb's law equation: Electric field (E) = (1 / (4πε₀))*(q / r²), where ε₀ is the permittivity of free space.
2. Determine at what distance from the ball the ball enters the field of the block to experience a force to just move it:
- The ball will experience a force when the electric field produced by the block is equal to the electric field required to overcome the forces holding the ball in place (e.g., gravitational force).
- Compare the electric field calculated from step 1 to the electric field required for the ball to move (considering the negligible mass of the ball).
Task b:
To find the expression for the velocity of the block that will enable the ball to rise to the top of the loop, you can use conservation of energy. At the top of the loop, the potential energy of the ball is equal to its initial kinetic energy.
1. Calculate the initial kinetic energy of the ball:
- Use the formula for kinetic energy: KE = (1/2)*m*v^2, where m is the mass of the ball and v is its velocity.
- The mass of the ball is negligible, so we can approximate it as zero.
2. Calculate the potential energy at the top of the loop:
- Use the formula for gravitational potential energy: PE = m*g*h, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height of the top of the loop.
3. Set the initial kinetic energy equal to the potential energy and solve for the velocity:
- (1/2)*m*v^2 = m*g*h
- Simplify and solve for v.
Task c:
To find the expression for the acceleration of the block that will enable the ball to rise to the top of the loop, you can use Newton's second law of motion.
1. Set up the equation of motion for the ball:
- The net force on the ball is the sum of the gravitational force and the force due to the electric field.
- Write down Newton's second law equation: F_net = m*a, where F_net is the net force, m is the mass of the ball, and a is its acceleration.
2. Express the forces on the ball in terms of acceleration:
- The gravitational force is m*g, where g is the acceleration due to gravity.
- The force due to the electric field is q*E, where q is the charge of the ball and E is the electric field.
3. Set up the equation F_net = m*a and solve for a.
Task d:
To calculate the charge and dimensions of the block for which the ball will be driven to the center of the semicircle without moving the block, you can use the concept of work done.
1. Calculate the work done by the electric field:
- The work done is equal to the product of the force and the displacement.
- Calculate the force using Coulomb's law and the displacement using the distance between the ball and the center of the semicircle.
2. Equate the work done by the electric field to the potential energy of the ball:
- The work done should be equal to the change in potential energy of the ball.
3. Solve the equation for the charge and dimensions of the block that satisfy the condition.