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At the beginning of the nineteenth century, the magnetic field of wires carrying currents was the focus of investigations in physics, both experimentally and theoretically. A particularly interesting case is that of a very long wire, carrying a constant current I, which has been bent into the form of a ‘V’, with opening angle 2θ. According to Ampere’s computations, the magnitude ` B of the magnetic field at a point P lying outside the ‘V’, but on its axis of symmetry and at a distance d from its vertex, is proportional to tan(θ/2). However, for the same situation, Biot and Savart suggested that the magnetic field at P might be proportional to θ. In fact, they attempted to decide between the two possibilities by measuring the oscillation period of a magnetic needle as a function of the ‘V’ opening angle. However, for a range of θ values, the predicted differences were too small to be measured. (i) Which formula might be correct? (ii) Find the proportionality factor in this formula and guess the most likely factor appearing in the other one.

Step-by-step

(i) When the angular opening approaches 2π, point P is in between the wires and arbitrarily close to them, and so both halves of the current-carrying wire produce very large magnetic fields at P, and in the same direction. Thus in this case the magnitude of the net magnetic field at P approaches infinity. As tan(θ/2) also approaches infinity when θ approaches π, Ampere’s formula may be correct. However, the expression is given by Biot and Savart must be wrong because it gives a finite value for B(P). In fact, Ampere’s result was later embodied in Maxwell’s electromagnetic theory and is now universally accepted.
(ii) When the angular opening is 2θ = π, the ‘V’ becomes a straight infinite wire. In this case, the magnitude of the field B(P) is known to be B = {{\mu }_{0}}I/(2πd). Since tan(θ/2) = tan(π/4) = 1, the proportionality factor in Ampere’s formula is {{\mu }_{0}}I/(2πd).
Biot and Savart chose their formula in such a way that it agreed with the expression for the magnetic field due to a straight infinite current-carrying wire already generally accepted. Thus they had as their proportionality factor {{\mu }_{0}}I/({{\pi }^{2}}d).
Note. In the region θ < π/2 the difference between the two predictions is relatively small. The ratio of the predicted values for B, 2θ/π tan(θ/2), shows the greatest difference from unity when θ → 0 and has a maximum value of 4/π.

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