First slide

Magnetic field due to current element

Question

Let two long parallel wires a distance ‘d’ apart, carry equal current ‘i’ in the same direction. One wire is at x=0, the other at x=d. Determine magnetic field at any point on x-axis between the wires as a function of ‘x’.

Moderate

A

$B = \frac{μ_{0} i (d - 2 x)}{2 π x (d - x)} \hat{j}$
B

$B = \frac{μ_{0} i (2 x - d)}{2 π d (d - x)} \hat{i}$
C

$B = \frac{μ_{0} i (x - 2 d)}{π x (d - x)} \hat{k}$
D

$B = \frac{μ_{0} i}{2 π d} (\frac{2 x - d}{d - x}) (- \hat{j})$

Solution

Let us consider a point ‘P’ at a distance $(x, y)$ from the wire 1.

$0 \leq x \leq d$
Hence ${\vec{B}}_{1} = \frac{μ_{0} i}{2 π x} \hat{j}$
The point is at a distance $d - x$ from wire 2.
$∴ {\vec{B}}_{2} = - \frac{μ_{0} i}{2 π (d - x)} \hat{j}$
$∴ {\vec{B}}_{n e t} = {\vec{B}}_{1} + {\vec{B}}_{2} = \frac{μ_{0} i}{2 π x} \hat{j} - \frac{μ_{0} i}{2 π (d - x)} \hat{j}$
${\vec{B}}_{n e t} = \frac{μ_{0} i (d - 2 x)}{2 π x (d - x)} \hat{j}$ .

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