A very long straight conductor and an isosceles triangular conductor lie in a plane and are separated from each other as shown in the figure. If a = 10 cm, b = 20 cm, and h = 10 cm find the coefficient of mutual induction.

Options

A. `2xx10^(-7)[1+ln2]`
B. `4xx10^(-8)[1+ln2]`
C. `4xx10^(-9)[1+ln2]`
D. `2xx10^(-10)[1+ln2]`

Answer

C

Explanation

Consider an elementary section of coil as shown in figure. Area of this section
`dA=ydx`
But `(y)/(b)=(x)/(h) rArr y=(b)/(h)x`
`dA=(b)/(h)xdx`
Magnetic flux linked with this section,
`d phi=BdA=(mu_(0h)l)/(2pi(a+x)).(b)/(h)xdx" "rArr" "d phi=(mu_(0h)//b)/(2pih).(x)/(a+x)dx`

`therefore" "` Total magnetic flux linked with the coil
`phi=(mu_(0h)lb)/(2pih)int_(0)^(h)(x)/(a+x)dx`
`=(mu_(0)lb)/(2pih)int_(0)^(h)(x)/(a+x)dx`
`=(mu_(0)lb)/(2pih)[x-a ln|a+x|]_(0)^(h)=(2xx10^(-7)xx20xx10^(-2))/(10xx10^(-2))" "[10xx10^(-2)+10xx10^(-2)ln|(2)/(1)|]`
`=4xx10^(-9)[1+ ln 2]H`