Question

A current `I(=4A)` flows along a thin wire PQRS shaped as shown in figure. The radius of the curved part of the wire is `10*0cm`. The angle `theta=90^@`. Find the magnitude of the total magnetic field at the point O.

Answer 1

Correct Answer - `2*68xx10^-5T`
Here, `I=4A`, `r=10*0cm=0*10cm`,
As `/_POS=90^@` and `PO=OS=r`,
so `/_OPS=/_OSP=45^@`
Angle subteded at O by curved wire PQRS
`=(3pi)/(2)rad`
Magnetic field at O due to current through portion SP of wire is
`B_1=(mu_0)/(4pi)(I)/(rcostheta//2)[sin theta/2+sin theta/2]`
`=(mu_0)/(4pi)I/rxx2tan theta/2`
It is acting normally downwards.
Magnetic field at O due to current through portion PQRS of wire is
`B_2=(mu_0)/(4pi)I/r[2pi-pi//2]=(mu_0)/(4pi)I/rxx(3pi)/(2)`
It is acting normally downwards.
Total magnetic field at point O is
`B=B_1+B_2=(mu_0)/(4pi)I/r[2 tan theta/2+(3pi)/(2)]`
`=10^-7xx(4)/(0*1)[2 tan 90^@/2+(3pi)/(2)]`
`=10^-7xx40[2 tan 45^@+(3pi)/(2)]`
`=10^-7xx40[2xx1+3/2pi]`
`=2*68xx10^-5T` acting normally downwards.