A circular loop of radius 0.3 cm lies parallel to amuch bigger circular loop of radius 20 cm. The centre of the small loop is on the axis of the bigger loop. The distance between their centres is 15 cm. If a current of 2.0 A flows through the smaller loop, then the flux linked with bigger loop is

Options

A. `6 xx 10^(-11)` weber
B. `3.3 xx 10^(-11)` weber
C. `6.6 xx 10^(-9)` weber
D. `9.1 xx 10^(-11)` weber

Answer

D

Explanation

`R=20 xx 10^(-2) m`, `r=0.3 xx 10^(-2)`

Magnetic field at centre of coil having radius R.
Magnetic field on axial line for coil 1 at distance x.
` B= (mu_(0)ir^(2))/(2(r^(2)+x^(2))^(3 // 2))`
Flux passes through coil 2
`phi_(21)=BA=BpiR^(2)(mu_(0)ir^(2)piR^(2))/(2(r^(2)+x^(2))^(3 // 2))`
`:.` Let current I be in bigger loop and flux through the smaller loop having radius r and at distance x =15 cm
`phi_(12)=B.a`
`phi_(12)=(mu_(0)IR^(2))/(2(R^(2)+x^(2)) xx pir^(2)`
Mutual inductance for coil 1 due to coil 2 (current following is I)
`phi_(12)=M_(12)I`
`M_(12)=(mu_(0)R^(2)pir^(2))/(2(R^(2)+x^(2))^(3 // 2))`
Flux linked through coil bigger coil.
`phi_(21)=M_(21)i=(mu_(0)R^(2)pir^(2)i)/(2(R^(2)+x^(2))^(3 // 2))`
`=(4pi xx 10^(-7) xx pi xx 4 xx 10^(-2) xx 9 xx 10^(6) xx 2)/(2(4 xx 10^(-2)+2.25 xx 10^(-2))^(3 // 2))`
`=9.1 xx 10^(-11) ` weber