First slide

Magnetic field due to current element

Question

Equal currents $i = 1 A$ are flowing through the wires parallel to y-axis located at $x = + 1 m,$ $x = + 2 m$ , $x = + 4 m$ and so on…,etc. but in alternate opposite directions as shown in figure. The magnetic field(in tesla) at origin would be $x \times 10^{- 7} \hat{k}$ , then the find the value of x.

Moderate

Solution

$F i e l d s o f w i r e s a t x = 1, 4, 16 a r e i n + z d i r e c t i o n w h e r e a s f i e l d s o f w i r e s a t x = 2, 8, 32 a r e i n - z d i r e c t i o n . \vec{B} = \frac{μ_{0} i}{2 π} [(\frac{1}{1} + \frac{1}{4} + \frac{1}{16} + ....) \hat{k} - (\frac{1}{2} + \frac{1}{8} + ....) \hat{k}] = \frac{μ_{0} i}{2 π} [(\frac{1}{1 - \frac{1}{4}}) \hat{k} - \frac{1}{2} (\frac{1}{1 - \frac{1}{4}}) \hat{k}] = \frac{2}{3} \frac{μ_{0} i}{2 π} \hat{k} = (\frac{2}{3} \times 2 \times 10^{- 7} \times 1) \hat{k} = 1.33 \times 10^{- 7} \hat{k}$

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