Two metallic rings `A` and `B` identical in shape and size but having different resistivities `rho_(A)` and `rho_(B)` are kept on top of two idential solenoids as shown in the figure.When current `I` is switched on in both the solenoids in identical manner, the rings `A` and `B` jump to heights `h_(A)` and `h_(B)` respectively with `h_(A)` > `h_(B)`. The possible relation(s) between their resistivities and their masses `m_(A)` and `m_(B)` is (are)

Options

A. `rho_(A) gt rho_(B)` and `m_(A)=m_(B)`
B. `rho_(A) lt rho_(B)` and `m_(A)=m_(B)`
C. `rho_(A) gt rho_(B)` and `m_(A)gtm_(B)`
D. `rho_(A) lt rho_(B)` and `m_(A)ltm_(B)`

Answer

B, D

Explanation

`F=B_(H)il`
`FDeltat=MV`
`i=(E_("induced"))/R`
`h=v^(2)/(2g)`
The horizontanl component of magnetic field due to solenoid will exert force on ring in vertical direction.
`F=B_(H)i(2pir)`
`FDeltat=MV`
`i=((Delta phi//Delta t))/((rho((2pir))/A))`
`B_(H)i(2pir)Delta t=MV`
`V=(B_(H)Delta phiA)/(rhoM)=K/(rhoM)`
`h=V^(2)/(2g)=K^(2)/(rho^(2)M^(2))`
`h_(A) gt h_(B)`
`K^(2)/(rho_(A)^(2)M_(A)^(2)) gt K^(2)/(rho_(B)^(2)M_(B)^(2))`
`rArr rho_(B)M_(B) gt rho_(A)M_(A)` to Using this we get