First slide

Magnetic field due to a straight current carrying wire

Question

Two parallel wires carrying equal currents in opposite directions are placed at x = ±a parallel to y-axis with z = 0. Magnetic field at origin O is B₁ and at P (2a, 0, 0) is B₂. Then the ratio B₁/B₂ is

Moderate

A

$3$
B

$\frac{1}{2}$
C

$\frac{1}{3}$
D

2

Solution

Case I: If the currents are in the directions shown in figure, then
${\vec{B}}_{1} = - 2 (\frac{μ_{0}}{2 π} \frac{i}{a}) \hat{k} = - \frac{μ_{0}}{πa} \hat{k} = magnetic field at O$
${\vec{B}}_{2} = \frac{μ_{0}}{2 π} \frac{i}{a} \hat{k} - \frac{μ_{0}}{2 π} \frac{i}{3 a} \hat{k} = magnetic field at P$
$\begin{array}{l} = \frac{μ_{0}}{3 π} \frac{i}{a} \hat{k} \\ ∴ B_{1} / B_{2} = 3 \end{array}$
Case II: In this case,
$\begin{array}{l} {\vec{B}}_{1} = \frac{μ_{0} i}{πa} \hat{k} \\ and {\vec{B}}_{2} = \frac{μ_{0} i}{3 πa} \frac{i}{a} \hat{k} \\ ∴ \frac{B_{1}}{B_{2}} = 3 \end{array}$

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