A long thin non-conducting cylindrical pipe having surface charge density sigma , radius R rotating with angular - Physics - Magnetism And Matter

Question

A long thin non-conducting cylindrical pipe having surface charge
density ‘sigma’, radius ‘R’ rotating with
angular velocity omega = kt about is axis, where k is a positive
constant Which of the following statement is/are correct for inside
region of pipe?
(A) Magnetic and electric field are uniform throughout region and
do not vary with time
(B) Magnetic field is constant throughout region but vary with time
and electric field is not constant throughout region and
independent of time
(C) Magnetic field is not constant throughout region but
independent of time and electric field is constant throughout
region and vary with time
(D) Energy density (u) due to magnetic field varies with time as u
= bt2 (Where b is positive constant)

Jay Sharma · Accepted Answer

Dear Student,

Surface Charge Density=σRadius =RCharge per unit length of cylinder, q=σ×πR2→For Magnetic field:Effective current through sruface of unit length of cylinder =qt=q2π/ω=qω2π=σπR2ω2πSo, Current I ∝ωWe know that: Magnetic field inside any current carrying ring or cylinder, B ∝ ISince,  I∝ω
 Hence, B ∝ω  Also, ω=kt
 So, B ∝t
→For Electric field:Here, Net charge is constant and uniform on pipe
 Charge is not changing with time
So, Electric field at any point inside pipe will also independent of time as there is no effect of moving charges on electric field
 From above two arguments, Option B is correct
→For Energy Density:Energy Density u=12εoE2+12B2μo=0+B22μo⇒u∝B2
 But, B ∝t
Hence, u∝t2 ⇒Option D is correct

Answer is B and D

Regards