First slide

Magnetic field due to a straight current carrying wire

Question

Determine the magnetic field at the centre of the current carrying wire arrangement shown in the figure. The arrangement extends to infinity. (The wires joining the successive squares are along the line passing through the centre)

Difficult

A

$\frac{μ_{0} i}{\sqrt{2} πa}$
B

0
C

$\frac{2 \sqrt{2} μ_{0} i}{πa} \ln 2$
D

none of these

Solution

Magnetic field at centre due to a square of side A:
$B = 4 [\frac{μ_{0}}{4 π} \frac{i}{(a / 2)} [\sin 45^{0} + \sin 45^{0}]] \frac{2 \sqrt{2} μ_{0} i}{πa}$
For the case in figure:
$\begin{array}{l} B_{net} = \frac{2 \sqrt{2} μ_{0} i}{π} [\frac{1}{a} - \frac{1}{2 a} + \frac{1}{3 a} - \frac{1}{4 a} + . ....] \\ = \frac{2 \sqrt{2} μ_{0} i}{πa} [1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + . ....] \\ = \frac{2 \sqrt{2} μ_{0} i}{πa} (\ln 2) \end{array}$

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