Question

A moving coil galvanometer has a coil with $175$ turns and area $1\ \text{c}{{\text{m}}^{2}}$. It uses a torsion band of torsion constant ${{10}^{-6}}\ \text{N}\cdot \text{m}\cdot \text{ra}{{\text{d}}^{-1}}$. The coil is placed in a magnetic field $B$ parallel to its plane. The coil deflects by $1{}^\circ $ for a current of $1\ \text{mA}$. The value of $B$ (in Tesla) is approximately
A. ${{10}^{-3}}$
B. ${{10}^{-1}}$
C. ${{10}^{-4}}$
D. ${{10}^{-2}}$

Answer 1

Hint: When a current carrying coil is put in a uniform magnetic field, it experiences a torque. At equilibrium, the deflecting couple must be equal to the restoring torque. Equate the two and determine the value of the applied magnetic field.
Formula Used:
\[{{\tau }_{deflecting}}=NIAB\]
\[{{\tau }_{restoring}}=C\theta \]

Complete answer:
A moving coil galvanometer is a highly sensitive device which is used to measure even small electric currents. It consists of a closed current carrying loop which experiences torque when placed in an external magnetic field. A restoring torque acts on the loop which must be equal to the deflecting couple so as to achieve equilibrium.
The deflecting couple that acts on the loop when placed in external magnetic field is given as,
\[{{\tau }_{deflecting}}=NIAB\]
Here, \[{{\tau }_{deflecting}}\] is deflecting couple, $N$ is number of turns, $I$ is current through the loop, $A$ is cross sectional area of loop and $B$ is magnitude of external field.
The restoring torque is given as,
\[{{\tau }_{restoring}}=C\theta \]
Here, \[{{\tau }_{restoring}}\] is restoring torque, $C$ is torsional constant and $\theta $ is deflection produced.
Now, at equilibrium,
\[{{\tau }_{deflecting}}={{\tau }_{restoring}}\]
  \[NIAB=C\theta \] … (1)
Now, the values provided to us include $N=175$, $A=1\ \text{c}{{\text{m}}^{2}}={{10}^{-4}}\ {{\text{m}}^{2}}$, $C={{10}^{-6}}\ \text{N}\cdot \text{m}\cdot \text{ra}{{\text{d}}^{-1}}$, $I={{10}^{-3}}\ \text{A}$ and $\theta =1{}^\circ =\dfrac{\pi }{180}\ \text{rad}$. Substitute these values in equation (1) to determine the magnitude of the magnetic field. The calculation can be seen as,
\[\begin{align}
  & NIAB=C\theta \\
 & {{10}^{-3}}\times {{10}^{-4}}\times 175\times B={{10}^{-6}}\times \dfrac{\pi }{180} \\
 & B={{10}^{-3}}\ \text{T}
\end{align}\]

Thus, the correct option is (A).

Note:
Convert all the given quantities in SI units before proceeding to the calculations. Convert the deflection produced from degrees to radians. Take care of the units in which the answer is required.