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Question: A short magnet of moment \[6.75A{m^2}\;\] produces a neutral point on its axis. If the horizontal co...

A short magnet of moment 6.75Am2  6.75A{m^2}\; produces a neutral point on its axis. If the horizontal component of the earth's magnetic field is 5×10−5Wb/m2  5 \times {10^{ - 5}}Wb/{m^2}\; then the distance of the neutral point should be
A. 10cm10cm
B. 20cm20cm
C. 30cm30cm
D. 40cm40cm

Explanation

Solution

An object’s tendency to align the magnetic with a magnetic field is called the magnetic moment or magnetic dipole moment. It can also be defined as the magnetic strength and orientation of a magnet or other objects that produce the magnetic field. It is a vector quantity. The magnet’s magnetic field is directly proportional to the magnetic moment.

Complete step by step solution:
The magnetic moment of the short magnet is given as m=6.75Am2  m = 6.75A{m^2}\;
It is given that the horizontal moment of the earth’s magnetic field is given has BH=5×10−5Wb/m2  {B_H} = 5 \times {10^{ - 5}}Wb/{m^2}\;
We know that the magnetic field formula is BH=μ04π  (2md3){B_H} = \dfrac{{{\mu _0}}}{{4\pi }}\;(\dfrac{{2m}}{{{d^3}}})
Here mm is the magnetic moment due to the magnet
ddis the distance from the neutral point’s position.
μ0{\mu _0} is the permeability of free space 10−7H/m{10^{ - 7}}H/m
Therefore substituting all the values.
5×10−5=  10−7×2×6.75d35 \times {10^{ - 5}} = \dfrac{{\;{{10}^{ - 7}} \times 2 \times 6.75}}{{{d^3}}}
d3=  10−7×2×6.755×10−5{d^3} = \dfrac{{\;{{10}^{ - 7}} \times 2 \times 6.75}}{{5 \times {{10}^{ - 5}}}}
d=(  10−7×2×6.755×10−5)1/3d = {(\dfrac{{\;{{10}^{ - 7}} \times 2 \times 6.75}}{{5 \times {{10}^{ - 5}}}})^{1/3}}
d=30cmd = 30cm
Therefore the correct option is C.

Note:
There is a theory that is believed to be the cause of Earth’s magnetic field. It is called the dynamo effect. This theory says that the magnetic field lines are caused by the presence of metallic fluids that are present in the inner core as well as the outer core of the Earth’s surface. The inner core consists of solidified elements and the outer core consists of molten iron. The earth’s magnetism is caused by the conventional current of nickel and molten iron. Only because of the presence of the magnetic field lines around the earth we have survived the solar wind coming out from the sun. If there was no magnetic shield the solar wind would have slowly destroyed our atmosphere. Mars does have a magnetic field protecting it.