4. A positively charged bead (point-like charge q) moves with no friction along

Question

4. A positively charged bead (point-like charge q) moves with no friction along a vertical rod, as sketched in Fig. 4, subject both to the Earth gravity and to the electric force due to another positive point-like charge Q fixed at the lower end of the rod. The bead is observed to execute an oscillating motion between "levels" A and B shown in the figure along with the relevant geometrical dimensions. Based on this observation, find mass m of the bead. The bead and fixed charge have small sizes that can be neglected in comparison with other dimensions. Figure 4: Illustration for problem 4.

Explanation / Answer

There are two positions where the electrostatic force is balanced by gravitational force.

Fe = Fg

KQ q / h2 = mgh ...............1

Kq Q / (h+d)2 = mg(h+d) ..............2

equ 1 - eq2

Kq Q ( 1/ h2 - 1/ (h + d)2 ) = mgh - mgh -mgd

KqQ / ( 1/h2 - 1 / (h +d)2 = -mgd

m = - (KqQ / gd ) (( h +d)2 - h2 )) / h2 ( h+d)2

m = (- KqQ / gd) (h2 +d2 -2hd -h2)) / h2 (h+d)2

m = (- kqQ / gd) ( d2-2hd) / h2 (h+d)2

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