A long, straight wire carries a current `i`. Let `B_1` be the magnetic field at a point P at a distance d from the wire. Consider a section of length `l` of this wire such that the point P lies on a perpendicular bisector of the section. Let `B_2` be the magnetic field at this point due to this section only. Find the value of d/l so that `B_2` differs from `B_1` by 1 %.

Explanation

We know that the magnetic field due to an infinitely long wire is given by the following relation:
` B_(1) = (mu_(0)i)/(2 pi d) ` …(i)
Now let us find that magnetic field due to l length on its perpendicular bisector. Here for perpendicular bisector,
` sin alpha = sin beta = (l//2)/sqrt(d^(2) + l^(2) //4) = l/sqrt(l^(2) + 4d^(2))`
Magnetic field can be written as follows:

`B_(2) = (mu_(0)i)/(4 pi d) (sin alpha + sin beta)`
` rArr" " B_(2) = (mu_(0)i)/(4 pi d) (2 xx l/sqrt(l^(2) + 4d^(2)))`
` rArr" " B_(2) = (mu_(0)i)/(2 pi d) (l/sqrt(l^(2) + 4d^(2)))` ....(ii)
Difference between `B_(1) " and " B_(2)" is " 2%` .
` rArr " " (B_(1) - B_(2))/B_(1) xx 100 = 2 `
` rArr " " 1 - B_(2)/B_(1) = 1/50`
Substituting the values in equations (i) and (ii) we get the following:
` rArr" " 1 - l/sqrt(l^(2) + 4d^(2)) = 1/50`
` rArr " " 1 - (1 + (4d^(2))/l^(2))^(-1//2) = 1/50`
` rArr " " 1 - (1 - (2d^(2))/l^(2)) approx 1/50`
` rArr" " d^(2)/l^(2) = 1/100`
` rArr " " d/l = 1/10`