Five long wires A, B, C, D and E, each carrying current I are arranged to form edges of a pentagonal prism as shown in figure. Each carries current out of the plane of paper.
(a) What will be magnetic induction at a point on the axis O? Axis is at a distance R from each wire.
(b) What will be the field if current in one of the wires (say A) is switched off?
(c) What if current in one of the wire (say) A is reversed?

Options

A. `0,(mu_(0)I)/(2piR),(mu_(0)I)/(piR)`
B. `(mu_(0)I)/(2piR),0,(mu_(0)I)/(piR)`
C. `(mu_(0)I)/(2piR),(mu_(0)I)/(piR),(mu_(0)I)/(piR)`
D. `(mu_(0)I)/(2piR),(mu_(0)I)/(2piR),(mu_(0)I)/(piR)`

Answer

1

Explanation

`vec(B)_(A),vec(B)_(B),vec(B)_(C),vec(B)_(E)` and `vec(B)_(E)` represent magnetci fields due to long wires `A,B,C,D` and `E (` each carrying current `I` out of the plane of paper ) at a point on the axis `O`.
Note that `vec(B)_(A)_|_OA,vec(B)_(B)_|_OB,vec(B)_(C)_|_OC, vec(B)_(D)_|_OD,vec(B)_(E)_|_OE`
`(a)` Magnetic induction at `O` due to all five current`-` carrying conductors, i.e.
`vec(B)_(R)=vec(B)_(A)+vec(B)_(B)+vec(B)_(C)+vec(B)_(D)+vec(B)_(E)`
`=vec(B)_(A)(vec(B)_(C)+vec(B)_(D))( :. vec(B)_(B)=-vec(B)_(E))`
`=vec(B)_(A)-vec(B)_(A)( :. vec(B)_(C)+vec(B)_(D)=vec(B)_(A))=vec(0)`
`(b)` Magnetic field at `O` due to four current carrying conductors `B,CltD` and `E (` as wire `A` is switched off `) , i.e.`
`vec(B)_(R)=vec(B)_(B)+vec(B)_(C)+vec(B)_(D)+vec(B)_(E)=(vec(B)_(C)+vec(B)_(D))`
`=-vec(B)_(A)(` as `vec(B)_(C)+vec(B)_(D)=-vec(B)_(A))`
or `|vec(B)_(R)|=B_(A)=(mu_(0)I)/(2piR)`
Perpendicular to `AO`, TOWARDS LEFT
`(C)` Magnetic field at `O` due to five current carrying conductors with current in wire `A` reversed, `i.e`
`vec(B)_(R)^('' )=-vec(B)_(A)+vec(B)_(B)+vec(B)_(C)+vec(B)_(D)+vec(B)_(E)`
`= -vec(B)_(A)+(vec(B)_(C)+vec(D)_(D))" "(` as `+vec(B)_(B)=-vec(B)_(E))`
`=-vec(B)_(A)-vec(B)_(A)=-2vec(B)_(A)`
or `|B_(R)^(')|=-vec(B)_(A)+vec(B)_(B)+vec(B)_(C)+vec(B)_(D)+vec(B)_(E)`
`=-vec(B)_(A)-vec(B)_(A)=-2vec(B)_(A)`
or `|vec(B)_(R)^(')|=-2vec(B)_(A)=2((mu_(0)I)/(2piR))=(mu_(0)I)/(piR)`
Perpendicular to `AO`, towards left.