Answer

Verified

420.9k+ views

**Hint:**Here, the Earth is in the form of a sphere and iron sphere is magnetized at the center of Earth. That is why, we have used the formula of volume of the sphere to find the fraction of the volume of Earth occupied by the sphere. Also, we have used the formula of dipole moment to calculate the radius of the iron sphere.

**Complete step by step answer:**

(a) As given in the question, the dipole moment of earth, ${\mu _{total}} = 8.0 \times {10^{22}}\,J{T^{ - 1}}$Also, the dipole moment of iron atom is, $\mu = 2.1 \times {10^{ - 23}}\,J{T^{ - 1}}$The mass of iron atom is given by

$

m = 56u = 56 \times 1.66 \times {10^{ - 27}}kg \\

\Rightarrow m = 9.30 \times {10^{ - 26}}kg $

When the magnetization of the sphere at the center of Earth is saturated, then the total dipole moment is given by ${\mu _{total}} = N\mu $. Now, the iron sphere that is at the center of Earth consists of $N$ iron atoms. Therefore, the mass of the iron sphere will be $Nm$. Now, the mass of the iron sphere will be due to the center of gravity acting on the sphere and is given by

$Nm = \dfrac{{4\pi \rho {R^3}}}{3}$

$ \Rightarrow \,N = \dfrac{{4\pi \rho {R^3}}}{{3m}}$

Putting this value in the value of dipole moment, we get

${\mu _{total}} = \dfrac{{4\pi \rho {R^3}}}{{3m}}\mu $

$ \Rightarrow \,R = {\left( {\dfrac{{3m{\mu _{total}}}}{{4\pi \rho \mu }}} \right)^{\dfrac{1}{3}}}$

$ \Rightarrow \,R = \left[ {\dfrac{{3\left( {9.30 \times {{10}^{ - 26}}kg} \right)\left( {8.0 \times {{10}^{22}}J{T^{ - 1}}} \right)}}{{4\pi \left( {14 \times {{10}^3}kg\,{m^{ - 3}}} \right)\left( {2.1 \times {{10}^{ - 23}}\,J{T^{ - 1}}} \right)}}} \right]$

$ \therefore \,R = 1.8 \times {10^5}m$

**Therefore, the radius of the iron sphere that is at the center of earth is $1.8 \times {10^5}m$.**

(b) Now, the volume of the iron sphere is given by

${V_s} = \dfrac{{4\pi }}{3}{R^3}$

$ \Rightarrow \,{V_s} = \dfrac{{4\pi }}{3}{\left( {1.8 \times {{10}^5}} \right)^3}$

$ \Rightarrow \,{V_s} = \dfrac{{4\pi }}{3}\left( {5.83 \times {{10}^{15}}} \right)$

$ \Rightarrow \,{V_s} = 2.53 \times {10^{16}}\,{m^3}$

Also, the volume of the earth’s sphere is given by

${V_e} = \dfrac{{4\pi }}{3}{r^3}$

$ \Rightarrow \,{V_e} = \dfrac{{4\pi }}{3}\left( {6.37 \times {{10}^6}} \right)$

$ \Rightarrow \,{V_e} = 1.08 \times {10^{21}}{m^3}$

Now the fraction of the volume of the Earth occupied by the sphere is given by

$V = \dfrac{{{V_s}}}{{{V_e}}}$

$ \Rightarrow \,V = \dfrac{{2.53 \times {{10}^{16}}{m^3}}}{{1.08 \times {{10}^{21}}{m^3}}}$

$ \therefore \,V = 2.3 \times {10^{ - 5}}$

**Therefore, the fraction of the Earth occupied by the iron sphere is $2.3 \times {10^{ - 5}}$.**

**Note:**Here, in the above solution, the mass of the sphere is taken as $Nm = \dfrac{{4\pi \rho {R^3}}}{3}$ . This is because we have taken the mass of the iron sphere in terms of density and volume as given below

$

M = \rho V \\

\Rightarrow \,Nm = \rho V $

Here, we have taken the mass of the iron sphere as $Nm$ because it contains $Nm$ .

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How do you graph the function fx 4x class 9 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What organs are located on the left side of your body class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell