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A coil of mass m and charge q moves wi...

A coil of mass `m ` and charge ` q` moves with a constant velocity `v` along the positive `x - direction` . It enters a region containing a uniform magnetic field `B` directed along the negative ` z- direction` , extending from ` x =a to x = b`. The minimum value of `v` required so that the paRTICLE CAN JUST ENTER THE REGION `XgtB` IS

A

` (mo NI)/(b)`

B

` (2mo NI)/(a)`

C

` (mo NI)/ 2(b - a)` in `(b)/(a)`

D

` (mo NI)/2(b - a)` in `(a)/(b)`

Text Solution

Verified by Experts

The correct Answer is:
C
Let us consider a thickness ` dx` of wire . Let it be at a distance `x` from the centre `O`
Number of turns per unit length ` = (N)/(b - a)`
:. Number of turns in thickness ` dx = (N)/(b-a) dx`
Small amount of magnetic field is produced at `O` due to thickness ` dx` of the wire .
` dB = (mu_(0))/(2) ( NI)/(b - a) (dx)/(x)`
On integrating , we get , ` B = int_(a)^(b) (mu_(0))/(2) (NI)/( b-a) (dx)/(x) = mu_(0)/(2) (NI)/(b-a)`
` int_(a)^(b) (dx)/(x) = (mu_(0))/(2) (NI)/( b-a) [ log_(e)x]_(a)^(b)`
` B = (mu_(0))/(2) (NI)/(b-a) log_(e)(b)/(a)`
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