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`

Explanation

`(a)(mu_(0)I)/(4r)((2r^(2)-a^(2))/(4r^(2)-a^(2)) )` to the left
`(b) (mu_(0)I)/(pir)((2r^(2)+a^(2))/(4r^(2)+a^(2)))` towards the top of the page

Current density `J=(1)/(pia^(2)-2pi((a)/(2))^(2))=(2I)/(pia^(2))`
Let us consider both the cavities are carrying equal and opposite currents with current density `J`.
Let `B_(1),B_(2)` and `B_(3)` be magnetic fields due to complete cylinder, upper and lower cavity respectively.
`(a)` At point `P_(1)`
`vec(B)_(1)=-(mu_(0))/(4pi).(2I_(1))/(r)hat(i)=-(mu_(0))/(4pi).(2Jxxpia^(2))/(r)hat(i)=-(mu_(0)I)/(pir)`
`vec(B)_(2)=(mu_(0))/(4pi).(2I_(2))/(a)hat(i)=(mu_(0))/(4pi).(2Jxxpi((a)/(2))^(2))/(r-(a)/(2))hat(i)`
`=-(mu_(0)I)/(4pi(r-(a)/(2)))hat(i)`
`vec(B)_(3)=(mu_(0))/(4pi).(2I_(3))/(r+(a)/(2))hat(i)=(mu_(0))/(4pi(r+(a)/(2)))hat(i)`
`vec(B)=vec(B)_(1)+vec(B)_(2)+vec(B)_(3)`
`=(mu_(0)I)/(4pi)[-(4)/(r)+(1)/(r-(a)/(2))+(1)/(r+(a)/(2))]hat(i)`
`vec(B)=(mu_(0)I)/(4pir)[(2r^(2)-a^(2))/(4r^(2)-a^(2))]hat(i)`
`:. (vec(B))=(mu_(0)I)/(4pir)[(2r^(2)-a^(2))/(4r^(2)-a^(2))],` towards left.
`(b)` At point `P_(2)`

`vec(B)_(1)=(mu_(0))/(4pi).(2I_(1))/(r)hat(j)=(mu_(0)I)/(pir)hat(j)`
`vec(B)_(2)=(mu_(0))/(4pi).(2I_(2))/(sqrt(r^(2)+(a^(2))/(4)))[sin theta hat(i)-cos theta hat(j)]`
`=(-mu_(0)I)/(2pi sqrt(4r^(2)+a^(2)))[sin thetahat(i)+cos theta hat(j)]`
`vec(B)_(3)=(mu_(0))/(4pi).(2I_(3))/(sqrt(r^(2)+(a^(2))/(4)))[sintheta hat(i)-cos theta hat(j)]`
`=(mu_(0)I)/(2pisqrt(4r^(2)+a^(2)))[sin theta hat(i)-cos theta hat(j)]`
`vec(B)=vec(B)_(1)+vec(B)_(2)+vec(B)_(3)=(mu_(0)I)/(2pi)[(2)/(r)-(2 cos theta)/(sqrt(4r^(2)+a^(2)))]hat(j)`
but `cos theta =(r)/(sqrt(r^(2)+(a^(2))/(4)))=(2r)/(sqrt(4r^(2)+a^(2)))`
`:. vec(B)=(mu_(0)I)/(2pi)[(2)/(r)-(4r)/(4r^(2)+a^(2))]hat(j)`
`=(mu_(0)I)/(4pir)[(2r^(2)+a^(2))/(4r^(2)+a^(2))]hat(j)`
`(vec(B))=(mu_(0)I)/(4pir)[(2r^(2)+a^(2))/(4r^(2)+a^(2))],` upwards.