Two conducting circular rings P and Q, radii r and 2r, roll uniformly in opposite directions on conducting surface S with centre-of-mass speeds 2v and v respectively. A uniform field B is perpendicular to the rings. The potential difference between their highest points is xBvr. Find x.
Physics · Mixed Chapter
JEE Mains

Question

Two conducting circular rings P and Q, radii r and 2r, roll uniformly in opposite directions on conducting surface S with centre-of-mass speeds 2v and v respectively. A uniform field B is perpendicular to the rings. The potential difference between their highest points is xBvr. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Potential difference along a rolling circular conductor: For a ring of radius a rolling without slipping with centre speed u, a point at height y above the surface moves horizontally at speed u·y/a. The local motional field magnitude along the vertical diameter is B(u·y/a). Step 2 — Integrate from contact to highest point: ΔV = ∫₀²ᵃ B(u·y/a)dy = (Bu/a)[y²/2]₀²ᵃ = 2Bua. Step 3 — Ring P (radius r, speed 2v): Potential of top relative to surface = 2B(2v)r = 4Bvr. Step 4 — Ring Q (radius 2…