Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A thin ring of radius cm carries a charge of nC, spread evenly. A small bead of mass g and charge slides without friction on a thin insulating rod along the axis of the ring. It is released from rest at a point cm from the centre. Gravity can be ignored. (a) What is the potential at the centre of the ring? (b) What is the potential at the release point? (c) What is the speed of the bead as it passes through the centre?
A point charge has , with , and potentials add as numbers. A ring has on its axis. A dipole has far away. A pair of charges has , and a group has the sum over all pairs.
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Find the two potentials. The bead is negative and the ring is positive, so the bead is pulled to the centre. It gains kinetic energy equal to the potential energy that it loses.
Step 1. The centre. . So .
Step 2. The release point. The distance to the ring is . .
Step 3. The energy. . At the release point, . At the centre, . The potential energy falls by J, and this is the kinetic energy at the centre.
Step 4. The speed. From we want , so .
Answer: V, V and m/s.
Check. The mass is kg. A negative bead goes towards the higher potential, and V is higher than V. It fits.
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