Question

The magnetic induction at a point 1${A^o}$ away from a proton measured along its axis of spin is (magnetic moment of the proton is $1.4 \times {10^{ - 26}}A{m^2}$).
$
  (a){\text{ 0}}{\text{.28mT}} \\
  (b){\text{ 28mT}} \\
  (c){\text{ 0}}{\text{.028mT}} \\
  (d){\text{ 2}}{\text{.8mT}} \\
$

Answer 1

Hint: In this question use the direct formula that the magnetic induction at any point on the axis of magnetic dipoles is given as $B = \dfrac{{{\mu _o}}}{{4\pi }} \times \dfrac{{2m}}{{{d^3}}}$. The basic conversion of $1{A^o} = 1 \times {10^{ - 10}}$ meter along with proper substitution of values in the direct formula will help approach the solution.

Complete step-by-step solution -

Given data:
Magnetic moment (m) of the proton = $1.4 \times {10^{ - 26}}A{m^2}$
Distance (d) of the point and the proton = 1${A^o}$
Now as we know that on the axis of magnetic dipoles, magnetic induction due to any point is given as
$B = \dfrac{{{\mu _o}}}{{4\pi }} \times \dfrac{{2m}}{{{d^3}}}$......................... (1), often measured in Tesla (T).
Where, B = magnetic induction measured in TESLA.
              ${\mu _o}$ = permeability in the air.
               m = magnetic moment of the proton.
               d = distance of the point and the proton.
Now as we know that ${\mu _o}$ is the permeability in the air and whose value is $4\pi \times {10^{ - 7}}$(H/m).
And we all know that $1{A^o} = 1 \times {10^{ - 10}}$ meter.
Therefore, d = $1 \times {10^{ - 10}}$meter.
Now substitute the values in equation (1) we have,
$ \Rightarrow B = \dfrac{{4\pi \times {{10}^{ - 7}}}}{{4\pi }} \times \dfrac{{2 \times \left( {1.4 \times {{10}^{ - 26}}} \right)}}{{{{\left( {1 \times {{10}^{ - 10}}} \right)}^3}}}$T
Now simplify this equation we have,
$ \Rightarrow B = {10^{ - 7}} \times \dfrac{{2.8 \times {{10}^{ - 26}}}}{{1 \times {{10}^{ - 30}}}} = 2.8 \times {10^{ - 7 - 26 + 30}}$
$ \Rightarrow B = 2.8 \times {10^{ - 3}}$T
Now as we know 1mT = 1$ \times {10^{ - 3}}$T.
$ \Rightarrow B = 2.8$mT.
So this is the required magnetic induction at a point 1${A^o}$ away from a proton measured along its axis of spin.
So this is the required answer.
Hence option (D) is the correct answer.

Note – Magnetic induction is general and has the definition that it refers to the process by which any object is actually magnetized whenever it is placed in an external magnetic field. This magnetizing property of any material depends upon the composition of the material. It is advised to remember the direct formula as it helps save a lot of time.