A wire of length ℓ is bent in the form of an equilateral triangle and carries an electric current i. (a) Find the magnetic field B at the centre.

(b) If the wire is bent in the form of a square, what would be the value of B at the centre?


(a)



Let M be the midpoint of equilateral ΔABC


∴


current is given to be i


Hence, in ΔAOB








The angles made by points B and C with centre M are respectively,


Hence separation of the point from the wire is given by,


Thus, the magnetic field induced due to current in wire BC is-






Now,
Net magnetic field at point M is given as Magnetic field due to wire BC + Magnetic field due to wire CA + Magnetic field due to wire AB
Since all wires are equal,



It is perpendicular to the plane in outward direction if current is anticlockwise and in an inward direction if the current is clockwise.


(b)



The angles made by B and C with centre M are


Distance between point from the wire,


Thus, the induced magnetic field due to electric current in wire BC is given by





Since all wires are equal,
Net magnetic field at point M = 4 × Magnetic field due to wire BC





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