Q: 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 ?

Solution :(a) Magnetic field due to one wire is $\displaystyle B = \frac{\mu_0}{4\pi}\frac{2 I}{R} $

Magnetic field at O due to all the five wires will be Zero .

(b) As total magnetic field is Zero . Hence magnetic field due to wire A = Magnetic field due to all four wires B , C , D & E

Hence Magnetic field due to all four wires will be $\displaystyle B = \frac{\mu_0 I}{2 \pi R} $ ; Towards Left as magnetic field due ti wire A is towards right .

(c) If current in wire A is reversed then net magnetic field at O will be

B = Magnetic field at O due to A + Magnetic field at O due to wire B , C , D & E

$\displaystyle B = \frac{\mu_0}{4\pi}\frac{2 I}{R} + \frac{\mu_0}{4\pi}\frac{2 I}{R} $ ; Towards Left .

$\displaystyle B = \frac{\mu_0 I}{\pi R} $ (Towards Left .)