Question

The magnet of a vibration magnetometer is heated so as to reduce its magnetic moment by 30%. The new periodic time of the magnetometer will be
A) Increases by 25%
B) Increases by 36%
C) Decreases by 25%
D) Increases by 36%

Answer 1

Hint: A device that is used to measure the direction, strength, and change of a magnetic field on a location (near or on Earth or it can be in space) is known as a magnetometer. Basically, it is used to measure magnetic intensity and fields. The time period of a magnetometer depends basically on
Moment of inertia of the magnet ‘$I$’
External magnetic field ‘$H$’
Magnetic Moment ‘$M$’
Consider these points and use the formula for the Time period of vibration of the magnetometer and hence find percentage.

Complete step by step answer:
Step I: Time Period is given by
T = \[2\pi \sqrt {\dfrac{I}{{MB}}} \]

Step II: Let $T_1$ be the time period of the magnetometer before heating and $T_2$ be the time period after heating.
And let $M_1$ and $M_2$ be their respective magnetic moments
$T_1$=\[2\pi \sqrt {\dfrac{I}{{{M_1}B}}} \] ……(i)
$T_2$= \[2\pi \sqrt {\dfrac{I}{{{M_2}B}}} \]……(ii)

Step III: Given in the question is
\[{M_2} = {M_1} - 36\% {M_1}\]
=\[{M_1} - \dfrac{{36}}{{100}}{M_1}\]
=\[\dfrac{{100{M_1} - 36{M_1}}}{{100}}\]
=\[\dfrac{{64}}{{100}}{M_1}\]

Step IV: Dividing (i) and (ii),
\[\dfrac{{{T_1}}}{{{T_2}}}\]= \[\sqrt {\dfrac{{64{M_1}}}{{100{M_1}}}} \]
\[\dfrac{{{T_1}}}{{{T_2}}}\]=\[\dfrac{8}{{10}}\]
\[{T_2} = 1.25{T_1}\]

Step V: From equation (ii),
\[{T_2}\]=\[2\pi \sqrt {\dfrac{I}{{{M_2}B}}} \] ……(iii)
Substitute value of $T_2$ in equation (iii),
\[1.25{T_1} = 2\pi \sqrt {\dfrac{I}{{{M_2}B}}} \]……(iv)
Step VI: Subtracting (i) from (iv),
\[1.25{T_1} - {T_1} = 2\pi \sqrt {\dfrac{I}{{{M_2}B}}} - 2\pi \sqrt {\dfrac{I}{{{M_2}B}}} \]
\[0.25{T_1} = 0\]

Step VII: Calculating percentage $T_1$=25%.
The new time period for the magnetometer will increase by 25%.

Option A is the correct answer.

Note:
Magnetometers are used for various purposes and they have their application in different fields. They are used in
- Detecting submarines
- to find deposits of iron in different geographical areas
- by treasure hunters to detect the metals deep inside the earth
- These days magnetometers are also used in electronic gadgets like smartphones in order to receive messages from varying magnetic fields by other nearest electromagnets.