Question

Two axial hollow conducting cylinder shells carry equal and opposite currents. Magnetic field due to a point outside both the shells is $ {\vec B_1} $ due to one shell and $ {\vec B_2} $ due to the other shell, then:
 $ \left( A \right){{\vec B}_1} = {{\vec B}_2} \\
  \left( B \right){{\vec B}_1} = - {{\vec B}_2} \\
  \left( C \right)Depend\,on\,radius\,of\,cylinders \\
  \left( D \right)Depend\,on\,distance\,from\,conductors \\ $

Answer 1

Hint :In order to solve this question, we are going to first see on what factors the magnetic field of a cylindrical hollow conducting shell does depends and then, on checking the options for the value or the conditions for the magnetic fields due to the hollow shells, the correct option is chosen.The magnetic field is given by
 $ \vec B = \dfrac{{{\mu _0}I}}{{2\pi r}} $

Complete Step By Step Answer:
It is given that the magnetic field due to one shell is $ {\vec B_1} $ due to one shell and $ {\vec B_2} $ due to another shell. Then the net magnetic field at a point, say the point is $ P $ , will depend on some factors depending upon the formula for the magnetic field by a conductor, carrying current $ I $ ,
Thus, the magnetic field is given by
 $ \vec B = \dfrac{{{\mu _0}I}}{{2\pi r}} $
So, as we are given that the two shells carry equal and opposite currents, thus, the only factor that affects the value of the field is the distance of the point from the respective shells, i.e... $ r $
Now, if the distance were equal, the this case,
 $ {\vec B_1} = - {\vec B_2} $
Would have been true,
While if the distance were equal in the opposite directions
Then, $ {\vec B_1} = {\vec B_2} $ would have been true.
As the point is outside the shells, so the dependence on the radius is also not possible.

Note :
The magnetic field inside this shell is going to be zero everywhere as this cylindrical shell is a hollow one. Secondly, the magnetic fields to be equal just having an equal and opposite current is not enough, it is also very important that the point at which the magnetic field needs to be determined is equal for both.