A point charge q=1 C is located at a distance 2 cm from one end of a thin insulating wire of length 10 cm having a charge Q=24 C , distributed uniformly along its length, as shown in figure. Force between q and wire is _ _ _ _ N. (Use : 1 4 _ o =9 10⁹ ~N . m ² / C ² )

Step-by-step solution

Consider a small element of length dx on the wire at a distance x from the point charge q . The linear charge density of the wire is = Q L = 24 10⁻⁶ 0.1 = 240 10⁻⁶ C/m. The charge on the small element is dQ = dx . The electrostatic force between the point charge q and the element dQ is dF = k q dQ x^2 = k q dx x^2 . The total force F is obtained by integrating dF from x = a to x = a + L , where a = 2 cm and L = 10 cm. F = _ 0.02 ^ 0.12 k q x^2 dx = k q [ - 1 x ]_ 0.02 ^ 0.12 F = k q ( 1 0.02 - 1 0.12 ) = k q Q L (