The figure shows a square loop PQRS with an edge length a . The resistance of the wire PQR is r and that of PSR is 2 r . The value of the magnetic field at the centre of the loop is found to be 2 μ 0

Question

The figure shows a square loop PQRS with an edge length a . The resistance of the wire PQR is r and that of PSR is 2 r . The value of the magnetic field at the centre of the loop is found to be 2 μ 0 i k π a , then what is the value of k ?

TestHub · Accepted Answer

The magnetic field due to a straight current-carrying wire is

$B = \frac{μ_{0} I}{4 π d} [\sin α + \sin β]$

$B_{PQR} = 2 \times \frac{μ_{0} (\frac{2 i}{3})}{4 π (\frac{a}{2})} [\sin 45 ° + \sin 45 °]$

$B_{PSR} = 2 \times \frac{μ_{0} (\frac{i}{3})}{4 π (\frac{a}{2})} [\sin 45 ° + \sin 45 °]$

$B_{net} = B_{PQR} - B_{PSR}$

$\Rightarrow B_{net} = \frac{\sqrt{2} μ_{0} i}{3 π a}$

$\Rightarrow k = 3$

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