Question

In a deflection magnetometer experiment, the deflections produced separately by two short bar magnets kept at the same distance is ${{45}^{o}}$ and ${{30}^{o}}$. Then the ratio of the magnetic moments of the two magnets is:
(A). $\sqrt{3}:2$
(B). $\sqrt{3}:1$
(C). $\sqrt{2}:1$
(D). $1:\sqrt{3}$

Answer 1

Hint: A magnetometer is an instrument which is used to measure the magnetic dipole moment and the magnetic field. It works on the principle of the tangent law which gives a relation between the horizontal component of Earth’s magnetic field and the magnetic moment of the magnet. Applying the tangent law for both magnets and dividing them, we can calculate the ratio of magnetic moments of the magnets

Formula used:
$B={{B}_{h}}\tan \theta $

Complete step-by-step answer:
A magnetometer is an instrument which consists of two long metre scales connected to a compass with two needles. It is used to measure the magnetic dipole moment or the magnetic field. It works on the principle of the tangent law which states that when a magnet is exposed to a magnetic field which is perpendicular to the horizontal component of the Earth’s magnetic field, then the magnet field will align such that it makes an angle $\theta $ . Therefore,
$B={{B}_{h}}\tan \theta $ - (1)
Here, $B$ is the magnetic moment of the magnet
${{B}_{h}}$ is the magnitude of the horizontal component of the Earth’s magnetic field
From eq (1), magnetic moment for the magnet that makes ${{45}^{o}}$ will be-
${{B}_{1}}={{B}_{h}}\tan 45$
$\Rightarrow {{B}_{1}}={{B}_{h}}$ - (2)
From eq (1), magnetic moment for the magnet that makes an angle ${{30}^{o}}$ will be-
${{B}_{2}}={{B}_{h}}\tan 30$
$\Rightarrow {{B}_{1}}=\dfrac{{{B}_{h}}}{\sqrt{3}}$ - (3)
Dividing eq (2) and eq (3), we get,
$\begin{align}
  & \dfrac{{{B}_{1}}}{{{B}_{2}}}=\dfrac{{{B}_{h}}}{\dfrac{{{B}_{h}}}{\sqrt{3}}} \\
 & \Rightarrow \dfrac{{{B}_{1}}}{{{B}_{2}}}=\sqrt{3} \\
\end{align}$
The ratio is $\sqrt{3}$
Therefore, the ratio of the magnetic moments of the two magnets is $\sqrt{3}:1$.

So, the correct answer is “Option B”.

Note: The magnetic moment of a magnet is the value of its strength and orientation. The magnet aligns itself according to the external magnetic field and horizontal component of the magnetic field which act on it. The needle of the compass is aligned along the Earth’s magnetic field.