Consider a wire carrying a steady current, I placed in a uniform magnetic field `vecB` perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do not work. This implies that

Options

A. Maximum possible magnetic field is `B_(0) + (mu_(0)I)/(2 pi R)`
B. Minimum magnetic field at some point is zero if `R lt (mu_(0)I)/(2 pi B_(0))`
C. Minimum magnetic field at some point is `B_(0) - (mu_(0)I)/(2 pi R) , " if " R gt (mu_(0)I)/(2 pi B_(0))`
D. Magnetic field at a point on its axis is `B_(0)`.

Answer

a, b, c, d

Explanation

Magnetic field lines due to the current in cylinder are circles perpendicular to the length of cylinder and hence at some points, its magnetic field will become parallel or anti-parallel to the applied magnetic field.
Maximum possible magnetic field is obtained for points where the field due to cylinder is maximum (points on its surface) and parallel to the applied magnetic field. Hence `B_(0) + (mu_(0) I)/(2 pi R)` will be the maximum possible magnetic field at some suitable point on its surface.
Option (a) is correct.
We know that at some point on its surface, the applied magnetic field is anti-parallel to the field created by current. But in case of the minimum possible magnetic field, there is a little twist.
Let us suppose that the applied magnetic field is less than the maximum field due to current, then there will be a point on either side of the surface where the field due to current is equal to the applied field and thus the resultant becomes zero for these points.
`B_(0) lt (mu_(0)I)/(2 pi R) rArr R lt (mu_(0)I)/(2 pi B_(0))`
Hence option (b) is correct.
If applied field is more than the maximum possible field due to current, then the minimum magnitude of magnetic field will be certainly on the surface and its magnitude will be `B_(0) - (mu_(0)I)/(2 pi R)` . This is the case
when `B_(0) gt (mu_(0)I)/(2 pi R) rArr R gt (mu_(0)I)/(2 pi B_(0))`
Option (c ) is correct.
Option (d) is also correct because magnetic field due to the current is zero at a point on the axis of the cylinder.