An infinitely long positively charged straight thread has a linear charge density λ Cm⁻¹. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is :

Solution

Physics sir · 4 steps

A line charge gives a field that falls as 1/r, and that 1/r cancels the r in mv²/r, so KE stays fixed.

1. Field from Gauss. Take a coaxial cylinder of radius r and length l as the Gaussian surface. Then E × 2π r l = (λ l)/ε₀, so E = λ/(2πε₀ r), pointing radially outward.
2. Force supplies centripetal. The electron is pulled towards the positive wire with force eE. This attraction is the centripetal force that keeps it on the circle.

   mv²/r = eλ/(2πε₀ r) ⇒ 1/2mv² = eλ/4πε₀

3. Look at r. The r on both sides cancels, so the kinetic energy eλ/4πε₀ has no r in it at all. It is the same for every radius.
4. So. The graph of KE against r is a horizontal straight line, which is option (d).

The idea to keep. For circular orbits, set the attractive force equal to mv²/r. If the force goes as 1/r, as for a line charge, KE is independent of radius. A 1/r² force gives KE proportional to 1/r.

Answer from the NTA final answer key. Solution by PrepWiser.