JEE Main 2023 · 24 January · evening · Single correct · +4 / −1
The electric potential at the centre of two concentric half rings of radii R₁ and R₂, having same linear charge density λ is :
- (a) λ/(2∈₀)
- (b) λ/(∈₀)
- (c) 2λ/(∈₀)
- (d) λ/(4∈₀)
Answer and solution
Answer (a) λ/(2∈₀)
Solution
Physics sir · 3 steps
Every bit of a half ring is at the same distance R from O, so its potential is just kq/R.
- One half ring. A half ring of radius R has length π R, so its charge is q = λ π R. Potential is a scalar, and every element of the ring sits exactly R from O, so you add up kq/R with no directions to worry about.
V_(half ring) = 1/4πε₀·(λ π R)/R = λ/4ε₀
- Radius drops out. The R cancels. A bigger ring carries more charge but is farther away by the same factor. So both half rings, of radii R₁ and R₂, give λ/4ε₀ each.
- So. V_O = λ/4ε₀ + λ/4ε₀ = λ/2ε₀, which is option (a).
The idea to keep. For any arc centred on the point, V = λθ/4πε₀, where θ is the angle the arc subtends. It does not depend on the radius, so concentric arcs just add their angles.
Answer from the NTA final answer key. Solution by PrepWiser.