GATE-2018 (24) :Wyllie – Time average equation

CSIR_NET_JUNE_2019

89) If the earth were to be twice its present size, its average density and magnetization remaining same, then which one of the following statements would be true with respect to earth’s gravity(g) and magnetic field(F) .

A) Both g and F are doubled

B) g is doubled but F remains the same

C) F is doubled but g remains the same

D) Both g and F remains the same

(Thanks to Chandrasekhar, ANU)

Solution:

Case(1):

Acceleration due to gravity $(g )= \frac{Gm}{r^{2}}$-----------------------------(1)

M= $density\times volume$

   =$\rho\times\frac{4}{3}\pi r^{3}$

Substitute this value in eq (1) 

 

$g = \frac{G\times\rho\times\frac{4}{3}\pi r^{3}}{r^{2}}$

$g = \frac{G\times\rho\times\frac{4}{3}\pi (2r)^{3}}{(2r)^{2}}$ 

since r= 2r

$g = \frac{G\times\rho\times\frac{4}{3}\pi r^{3}\times8}{r^{2}\times4}$

$g = 2 ( \frac{G\times\rho\times\frac{4}{3}\pi r^{3}}{r^{2}})$

$g=2g$

Case(2):

Magnetic Field of the earth

$(F )= \frac{\mu_{0}m\sqrt{1+3sin\phi^{2}}}{4\pi r^{3}}$ 

$magnetic momoent (m) = I\times V$

                                      = $I\times \frac{4}{3}\pi r^{3}$

$(F )= \frac{\mu_{0}m\sqrt{1+3sin\phi^{2}}}{4\pi r^{3}}$

=$\frac{\mu_{0}\times I\times \frac{4}{3}\pi r^{3}\sqrt{1+3sin\phi^{2}}}{4\pi r^{3}}$

=$\frac{\mu_{0}\times I\times \frac{4}{3}\pi (2r)^{3}\sqrt{1+3sin\phi^{2}}}{4\pi (2r)^{3}}$ 

since r=2r

=$\frac{\mu_{0}\times I\times \frac{4}{3}\pi (2r)^{3}\sqrt{1+3sin\phi^{2}}}{4\pi (2r)^{3}}$ 

 

=$\frac{\mu_{0}\times I\times \frac{4}{3}\pi 8r^{3}\sqrt{1+3sin\phi^{2}}}{4\pi 8r^{3}}$

=$\frac{\mu_{0}\times I\times \frac{4}{3}\pi r^{3}\sqrt{1+3sin\phi^{2}}}{4\pi r^{3}}$

=F

F= F      if r= 2r.

If the earth size  were doubled then its present, g is doubled but F remains same