(i) A pair of stationary and infinitely long bent wires are placed in the XY- plane as shown.The wires carry currents 10 A each. The segment P and Q are parallel to the Y-axis such as OS =OR = 0.02 m. Find the magnitude field at the origin O

(ii) Three long wires carrying currents `10 A` ,`20 A` and `30 A` are placed parallel to each other as shown. Find the magnetic field at P and Q

Explanation

(a) (i)
At `P_(1):`
`B_(1)=(mu_(0)(10))/(2pi(2))=(5mu_(0))/(2pi), o.`
`B_(2)=(mu_(0)(20))/(2pi(2))=(10mu_(0))/(2pi), ox`
`B_(P_(1))=B_(2)-B_(1)=(5mu_(0))/(2pi),ox`
(ii)
At `P_(2):`
`B_(1)=(mu_(0)(10))/(2pi(2))=(5mu_(0))/(2pi), o.`
`B_(2)=(mu_(0)(20))/(2pi(2))=(10mu_(0))/(2pi), o.`
`B_(P_(2))=B_(1)-B_(2)=(15mu_(0))/(2pi),o`.
(iii)
At `P_(3):`
`B_(1)=(mu_(0)(10))/(2pi(2))=(5mu_(0))/(2pi), ox`
`B_(2)=(mu_(0)(20))/(2pi(2))=(10mu_(0))/(2pi), o.`
`B_(P_(3))=B_(2)-B_(1)=(5mu_(0))/(2pi),o.`
(iv)
At `P_(4):`
`B_(1)=(mu_(0)(10))/(2pi(2))=(5mu_(0))/(2pi), ox`
`B_(2)=(mu_(0)(20))/(2pi(2))=(10mu_(0))/(2pi), ox`
`B_(P_(4))=B_(1)+B_(2)=(15mu_(0))/(2pi),ox`
(b)
`B_(P)=(mu_(0)i_(1))/(2pid)-(mu_(0)i_(2))/(2pid)=(mu_(0))/(2pid)(i_(1)-i_(2))=20`

`B'_(P)=(mu_(0)i_(1))/(2pid)-(mu_(0)i_(2))/(2pid)=(mu_(0))/(2pid)(i_(1)+i_(2))=80`
`(i)//(ii)` `(i_(1)-i_(2))/(i_(1)+i_(2))=20/80=1/4`
`4i_(1)-4i_(2)=i_(1)+i_(2)`
`3i_(1)=5i_(2)`
`(i_(1))/(i_(2))=5/3`
(c)
At `P:`
`B_(1)=(mu_(0)i)/(2piy), o.`
`B_(2)=(mu_(0).2i)/(2pix), ox`
Since magnetic field at `P` is zero
`B_(1)=B_(2)`
`(mu_(0)i)/(2piy)=(2mu_(0)i)/(2pix)`
`x=2y`
`2y-x=0`
Locus is a straight line passing through origin

(d)
`B_(1)=B_(4)=0`
`B_(2)=B_(3)=(mu_(0)i)/(4pid), o.`
`B_(O)=B_(2)+B_(3)=(mu_(0)i)/(2pid)=(mu_(0))/(2pi)xx10/0.02=(mu_(0))/(4pi)xx1000`
`=10^(-7)xx10^(3)=10^(-4)T`
(e)
At `Q:`
`B_(Q)=(mu_(0))/(2pi)[30/d-20/d-10/(3d)]`
`=(mu_(0))/(2pi).20/(3d) ox.`
At `P:`
`B_(P)=(mu_(0))/(2pi)[30/(3d)+20/d-10/(d)]`
`=(mu_(0))/(2pi).20/(d)`
`(B_(P))/(B_(Q))=3`