Since the wire is of circular shape hence, area of the circular loop is given as
A = πr
2
-------------- (2)
Where
r is the radius of the wire;
Let A
1
and r
1
be the area and radius of the initial wire. Therefore, expressing equation 2 in terms of
A
1
and r
1
we get
A
1
= πr
1
2
----------- (3)
Similarly, let A
2
and r
2
be the area and radius of the final wire. Therefore, expressing equation 2 in
terms of A
2
and r
2
we get
A
2
= πr
2
2
------------ (4)
Since it is given in the question radius of the final wire is twice that of the initial wire hence, r
2
= 2r
1
Using the value of r
2
in equation 4 we get
A
2
= π( 2r
1
)
2
Or, A
2
= 4πr
1
2
------------- (5)
Comparing equation 5 with 3 we find that final area can be expressed in terms of initial area and is
given as
A
2
= 4πr
1
2
A
2
= 4A
1
------------------ (6)
Expressing initial and final magnetic moment in equation 1 in terms of initial area and final area we
get
M
1
= IA
1
---------------- (7)
And, M
2
= IA
2
---------- (8)
Both initial and final current in the wire is same. Dividing equation 8 by equation 7 we get
M
2
/ M
1
= (IA
2
)/(IA
1
) ---------------- (9)
Cancelling out the common term, in 9 we get
M
2
/ M
1
= A
2
/A
1
--------------- (10)
Using equation 6 in 10 we get
M
2
/ M
1
= 4A
2
/A
1
Or, M
2
/ M
1
= 4
Therefore, M
2
= 4 M
1
-------------- (11)
Putting the value of M
1
in equation 11 we get M
2
= 4 * 3 * 10
-5
Or, M
2
= 1.2 * 10
-4
Am
2
Hence, E is the correct answer option.
