NTA Abhyas JEE Main2020PhysicsElectrostaticsPractice
Two long coaxial cylindrical metal tubes (inner radius a , outer radius b ) stand vertically in a tank of dielectric oil (of mass density ρ , dielectric constant K ). The inner one is maintained at potential V and the outer one is grounded. To what equilibrium height ( h ) does the oil rise in the space between the tubes? [Assume this height ( h ) as an equilibrium height]
Options
- Aε 0 2 V 2 K - 1 gρ b 2 - a 2   ln b a
- Bε 0 V 2 K - 1 ρ b 2 - a 2 g   ln b a
- C4 ε 0 V 2 K - 1 gρ b 2 - a 2 ln b a
- D6 ε 0 V 2 K - 1 ρ b 2 - a 2 g   ln b a
Correct answer
B. ε 0 V 2 K - 1 ρ b 2 - a 2 g   ln b a
Step-by-step solution
G r a v i t a t i o n a l   f o r c e = m g =   ρ π b 2 -   a 2 g h Net upward force F   = d U d h = 1 2 V 2   d C d h where h is the height of liquid. Now we calculate C as a function of h C e q   =   C A i r   +   C K = 2 π ε 0 L - h l n b a + 2 π ε 0 K h l n b a C   =   2 π ε 0 L + K - 1 h l n b a F   =   1 2 V 2 × d C d h   = 1 2 V 2 × 2 π ε 0 K - 1 l n b a ∴   F   =