A long straight wire in the horizontal plane carries a current of 75A in north to south direction, magnitude and direction of field B at a point 3m east of the wire is

A.)$4 \times {10^6}T,$ vertical up
B.)$5 \times {10^6}T,$ vertical down
C.)$5 \times {10^{ - 6}}T,$ vertical up
D.)$4 \times {10^6}T,$ vertical down

Hint- In this question we will use the ampere's circuital law which will give the value of required magnitude of the magnetic field and for the direction we will use screw rule or right hand rule which are stated as follows:

Complete step-by-step answer:
Ampere's circuital law- Ampere's circuital law states that the closed line integral of magnetic field around a current carrying conductor is equal to absolute permeability times the total current flowing through the conductor.

The Ampere's law mathematical formula is,

\[\oint B.dl = {\mu _0}I\]

Right hand screw law-The right hand rule specifies that: to evaluate the position of the magnetic force on a positive moving target, the right hand thumb pointing in the position of v, the fingers pointing in the direction of B, and the perpendicular to the palm points pointing in the direction of F.

Given

Current in the wire I = 75A
Distance from the wire R = 3m

As we know that the ampere’s circuital law gives the magnetic field of an enclosed path
Using ampere circuital law

\[\oint B.dl = {\mu _0}I\]
$B \times 2\pi R = {\mu _0}I$
Substituting the value of R and I in the above equation

$
  B = \dfrac{{{\mu _0} \times 75}}{{2\pi \times 3}} = 2 \times {10^{ - 7}} \times 25 \\
  B = 5 \times {10^{ - 6}}T \\
$

The direction of the field is determined using the right hand thumb rule. And according to this rule the direction of the magnetic field at point 3m away east is vertical upwards.
Hence, the correct option is B.

Note- In order to solve these types of questions, remember the ampere circuital law and similar other law’s like biot- savart law etc. Ampere’s circuital law is very important and has many applications and can be used to know what magnetic field is generated by an electric current. This is useful in building electromagnets, motors, generators, transformers, and more.