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Question: The magnetic induction at a distance of \(20\,cm\) from centre of a short magnetic dipole having mag...

The magnetic induction at a distance of 20 cm20\,cm from centre of a short magnetic dipole having magnetic moment 0.5 Am20.5\,A{m^2} on its axis is:
A. 1.57×10−5T1.57 \times {10^{ - 5}}T
B. 1.25×10−5T1.25 \times {10^{ - 5}}T
C. 1.57×10−4T1.57 \times {10^{ - 4}}T
D. 12.5×10−5T12.5 \times {10^{ - 5}}T

Explanation

Solution

In magnetism, as we know that magnetic moment is the strength of magnetic field produced by any material when placed in the presence of some magnetic field and magnetic induction is the magnitude of magnetic field produced by a body. We will use the basic relation between magnetic moment and magnetic induction to calculate the magnitude of magnetic induction.

Formula used:
If BB is the magnetic field at a distance of dd on axial line of a short magnetic dipole having magnetic moment of mm then
B=μ02m4πd3B = \dfrac{{{\mu _0}2m}}{{4\pi {d^3}}}
where, μ04π=10−7T\dfrac{{{\mu _0}}}{{4\pi }} = {10^{ - 7}}T.

Complete step by step answer:
According to the question, we have given that the magnetic moment at a distance of d=20cm=0.2md = 20cm = 0.2m is m=0.5Am2m = 0.5A{m^2} then, let B be the magnetic induction on axial line at given distance, then using the formula B=μ02m4πd3B = \dfrac{{{\mu _0}2m}}{{4\pi {d^3}}} and putting the values of given parameters we get,
B=μ02m4πd3B = \dfrac{{{\mu _0}2m}}{{4\pi {d^3}}}
⇒B=10−7×2×0.5(0.2)3\Rightarrow B = \dfrac{{{{10}^{ - 7}} \times 2 \times 0.5}}{{{{(0.2)}^3}}}
On solving this calculation we get,
B=10−70.008B = \dfrac{{{{10}^{ - 7}}}}{{0.008}}
∴B=1.25×10−5 T\therefore B = 1.25 \times {10^{ - 5}}\,T
So, the magnetic induction at a distance of 20 cm20\,cm from centre of a short magnetic dipole having magnetic moment 0.5Am20.5A{m^2} on its axis is B=1.25×10−5 TB = 1.25 \times {10^{ - 5}}\,T.

Hence, the correct option is B.

Note: It should be remembered that, the basic unit of conversion is used in solving the question is 1 cm=0.001 m1\,cm = 0.001\,m and μ0{\mu _0} is known as the permeability of free space which has a fixed value of 4π×10−7T4\pi \times {10^{ - 7}}T and here unit of magnetic induction is called Tesla denoted by TT and also magnetic moment is simply the current flowing in given surface area through a conductor.