A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?

Explanation

1. Magnetic moment of given coil is,
m = NIA
= `(1)(12)(10xx10^(-2)xx5xx10^(-2))`
`thereforem=0.06"(ampere)"("meter")^(2)`
2. Here B = 3000 G (Gauss) = `3000xx10^(-4)T=0.3T`
3. (a) Here, `vecm=0.06hatiandvecB=0.3hatk`
Now, `vectau=vecmxxvecB`
= `0.06hatixx0.3hatk`
`thereforevectau=-0.018hatj" "(becausehatixxhatk=-hatj)`
`thereforetau=0.018Nm` (Along - Y-axis)
(b) `vectau=vecmxxvecB`
= `0.06hatixx0.3hatk`
`thereforevectau=-0.018hatj" "(becausehatixxhatk=-hatj)`
`thereforetau=0.018Nm` (Along-Y-axis)
( c) `vectau=vecmxxvecB`
= `0.06(-hatj)xx0.3hatk`
= `-0.018hati" "(becausehatjxxhatk=hati)`
`thereforetau=0.018Nm` (Along -X-axis)
(d)
Here, `tau=mBsintheta`
= `(0.06)(0.3)sin90^(@)`
`thereforetau=0.018Nm`
Direction of `vectau` makes angle `30^(@)` clockwise with -Y-axis in the XY-plane.
(e) `vectau=vecmxxvecB`
= `0.06hatkxx0.3hatk`
= `vec0" "(becausehatkxxhatk=vec0)`
`thereforetau=0`
(f) `vectau=vecmxxvecB`
= `0.06(-hatk)xx0.3hatk`
= `vec0" "(becausehatkxxhatk=vec0)`
`thereforetau=0`
4. Here, magnetic forces exerted on the opposite sides of rectangular loop are colinear equal in magnitude and opposite in directions and so net force on the loop is zero in each case.
5. In case (e), we have `vecm||vecB` and so angle `theta` between them is `0^(@)` and so loop will be in the state of stable equailibrium because,
`U=-mBcostheta=-mBcos0^(@)=-mB` = min