Using the formula \[\mathop F\limits^ \to {\text{ }} = q\overrightarrow v {\text{ }} \times \overrightarrow B \] and \[B = \dfrac{{{\mu _0}i}}{{2\pi r}}\], show that the SI units of the magnetic field B and the permeability constant $\mu _0$ may be written as \[N{\text{ }}m{A^ - }^1\] and \[N{A^ - }^2\] respectively.

Hint:The International System of Units (abbreviated SI from system international in French) is a scientific way of representing the magnitudes or quantities of significant natural events. All SI units can be stated directly as well as in terms of conventional multiples or fractional quantities.

Complete step by step answer:
Lorentz force is defined as the result of electromagnetic fields combining magnetic and electric forces on a point charge. It is also known as the electromagnetic force and is used in electromagnetism. Lorentz force describes the mathematical formulae as well as the physical significance of forces acting on charged particles travelling through space with both an electric and magnetic field. This is why the Lorentz force is so important.

Step 1: Lorentz Force Relation:
We know that
\[\mathop F\limits^ \to {\text{ }} = q\overrightarrow v {\text{ }} \times \overrightarrow B \]
We get
$B = \dfrac{F}{{qv}} = \dfrac{F}{{\left( {1tv} \right)}}$
Units of
Force \[\left( F \right)\] = \[N\]
Current \[\left( I \right)\] = \[A\]
Time \[\left( T \right)\] = \[s\]
Velocity \[\left( v \right)\] = \[m/s\]

Step 2: Magnetic Field \[B\]
The magnetic force on moving electric charges, electric currents, and magnetic materials is described by a magnetic field, which is a vector field. In a magnetic field, a moving charge experiences a force that is perpendicular to both its own velocity and the magnetic field.
$B = \dfrac{N}{{A - m}}$
Now, a magnetic field formed by a continuous current running via a very long straight line encircles the wire. At a radial distance of \[P\] from the wire, it has magnitude.
$B = \dfrac{N}{{A - m}}$

Step3: Permeability constant
When producing a magnetic field in a classical vacuum, the permeability constant \[{\mu _0}\] (also known as the magnetic constant or the permeability of free space) is the proportionality between magnetic induction and magnetising force.
\[{\mu _0} = B\dfrac{{2\pi r}}{i} = \dfrac{N}{{A - m}} \times \dfrac{m}{A} \\
\therefore {\mu _0} = \dfrac{N}{{{A^2}}} \]

Note: SI unit names are usually written in lowercase in writing. The insignia of units named after people, on the other hand, are capitalised (e.g., ampere and \[A\] ). Periods are not required because these symbols are not abbreviations. Except for the degree symbol, a space should always be placed between a number and the SI unit. With SI units, italics are rarely utilised.