A long cylindrical conductor of radius a has two cylindrical cavities of diameter a through its entire length as shown in cross-section in figure. A current `I` is directed out of the page and is uniform throughout the cross-section of the conductor. Find the magnitude and direction of the magnetic field in terms of `mu_0,I,r` and a.

(a) at point `P_1` and (b) at point `P_2`

Answer

A, B, D

Explanation

Current per unit area
`sigma =I/(pia^w2-pi(a/2)^2-(a/2)^2)`
`(2I)/(pia^2)`
Total area is `(pia^2)`.Therefore the total current is
`I_1=(sigma)(pia^2)=2I`
Cavity area is `pi(a/2)^2` Therefore cavity current is
`I_2=(sigm)(pia^2/4)=I/2`
Now the given current system can be assumed as shown below.

a. At `P_1, B_1=mu_0/(2pi) (2I)/t=(mu_0I)/(pir) ("towards left")`
`B_2=mu_0/(2pi) (I/2)/((r-a/2))` (towards right)
and `B_2=mu_0/(2pi)(I/2)/((r+a/2))` (towards right)
`:. B_(net)=B_1-B_2-B_2` (toward left)
`=(mu_0I)/ pi [1/r-1/(4r-2a)-1/(4r+2a)]`
`=(mu_0I)/pi[(16r^3-4a^2-4r^2-2ar-4r^2+2ar)/(r(16r^2-4a^2))]`
`=(mu_0I)/(pir) [(2r^2-a^2)/(4r^2-a^2)]` (towards left)
b.

`B_(net)=B_1-2B_2costheta]` (towards the top)
`=mu_0/(2pi) (2I)/R-2[mu_0/(2pi) (I/2)/sqrt((r^2+a^2/4))]r/sqrt((r^2+a^2/4))`
`(mu_0I)/(pir) [(2r^2+a^2)/(4r^2+a^2)]`(towards the top)