Two tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of $60^{\circ}$ and $45^{\circ}$ respectively. The ratio of the number of turns in the coils is
A. $\quad 4 / 3$
B. $\dfrac{\sqrt{3}+1}{1} \\ $
C. $\dfrac{\sqrt{3}+1}{\sqrt{3}-1} \\ $
D. $\dfrac{\sqrt{3}}{1}$

Hint: To solve this problem, use the current in tangent galvanometer formula. Use the same formula to calculate the current in tangent galvanometers. Because these galvanometers are connected in parallel, the same current will flow through both of them. As a result, compare the equations to obtain an expression. Find the number of turns in tangent galvanometer B by substituting the given values into this obtained expression.

Formula used :
\[I = \dfrac{{2rH}}{{{\mu _0}N}}\tan \theta \]
Where,
r is the radius of the coil
H is the magnetic induction
N is the number of turns the coil has.

Complete step by step solution:
Tangent galvanometers were the first instruments used to measure small electric currents. It is made up of a coil of insulated copper wire wound around a nonmagnetic circular frame. Its operation is based on the principle of magnetism's tangent law. A magnetic field (B) is produced at the center of the circular coil when a current is passed through it in a direction perpendicular to the plane of the coil. The TG is configured in such a way that the horizontal component of the earth's magnetic field (BH) is directed in the direction of the coil's plane.

Two mutually perpendicular fields then act on the magnetic needle. If \[\theta \] is the needle's deflection then
\[{\rm{B}} = {{\rm{B}}_{\rm{h}}}\tan \theta \]
Where, \[B = \dfrac{{{\mu _0}{\rm{nI}}}}{{2{\rm{a}}}}\]
Where\[{\rm{n}}\] is the number of coils,\[{\rm{I}}\] is the current, and \[{\rm{a}}\] is the radius of the coil.

Given that the radius of both coils is the same. Because both coils are connected in series, the current will be the same.
\[{{\rm{B}}_1} = \dfrac{{{\mu _{\rm{o}}}{{\rm{n}}_1}{\rm{I}}}}{{2{\rm{a}}}} = {{\rm{B}}_{\rm{h}}}\tan {\theta _1} \ldots .({\rm{i}}) \\ \]
\[\Rightarrow {{\rm{B}}_2} = \dfrac{{{\mu _{\rm{o}}}{{\rm{n}}_2}{\rm{I}}}}{{2{\rm{a}}}} = {{\rm{B}}_{\rm{h}}}\tan {\theta _2} \ldots {\rm{.(ii) }} \\ \]
Divide both the expressions, we obtain
\[\dfrac{{{{\rm{B}}_1}}}{{\;{{\rm{B}}_2}}} = \dfrac{{\tan {\theta _1}}}{{\tan {\theta _2}}} \\ \]
This can be also written as,
\[\dfrac{{{{\rm{n}}_1}}}{{{{\rm{n}}_2}}} = \dfrac{{\tan {\theta _1}}}{{\tan {\theta _2}}} \\ \]
Substitute the values of \[\theta \] as given in the question:
\[\dfrac{{{{\rm{n}}_1}}}{{{{\rm{n}}_2}}} = \dfrac{{\tan 60}}{{\tan 45}} \\ \]
Simplify the values according to trigonometry identities:
\[\dfrac{{{{\rm{n}}_1}}}{{{{\rm{n}}_2}}} = \dfrac{{\sqrt 3 }}{1}\]
Therefore, the ratio of the number of turns in the coils is \[\dfrac{{\sqrt 3 }}{1}\].

Hence, the option D is correct.

Note: Students must understand the tangent galvanometer in order to answer this question. To increase the sensitivity of a tangent galvanometer, increase the number of turns of the coil or decrease the radius of the coil or the magnetic induction. Students must understand that as the number of turns of the coil increases, the radius of the coil does not remain constant across all turns and the magnetic induction does not become uniform at the center.