Question
An arrangement of three parallel straight wires placed perpendicular to plane of paper carrying same current $$I$$ along the same direction is shown in figure. Magnitude of force per unit length on the middle wire $$B$$ is given by
An arrangement of three parallel straight wires placed perpendicular to plane of paper carrying same current $$I$$ along the same direction is shown in figure. Magnitude of force per unit length on the middle wire $$B$$ is given by
A.
$$\frac{{{\mu _0}{i^2}}}{{2\pi d}}$$
B.
$$\frac{{2{\mu _0}{i^2}}}{{\pi d}}$$
C.
$$\frac{{\sqrt 2 {\mu _0}{i^2}}}{{\pi d}}$$
D.
$$\frac{{{\mu _0}{i^2}}}{{\sqrt 2 \pi d}}$$
Answer :
$$\frac{{{\mu _0}{i^2}}}{{\sqrt 2 \pi d}}$$
Solution :
As force on wire $$B$$ due to $$A$$ and $$C$$ are attractive, so we have following condition
$$F = \frac{{{\mu _0}{I^2}}}{{2\pi d}}$$
Resultant force on $$B$$
$$ = \sqrt {F_1^2 + F_2^2} = \sqrt 2 F = \sqrt 2 \times \frac{{{\mu _0}{I^2}}}{{2\pi d}} = \frac{{{\mu _0}{I^2}}}{{\sqrt 2 \pi d}}$$
As force on wire $$B$$ due to $$A$$ and $$C$$ are attractive, so we have following condition
$$F = \frac{{{\mu _0}{I^2}}}{{2\pi d}}$$
Resultant force on $$B$$
$$ = \sqrt {F_1^2 + F_2^2} = \sqrt 2 F = \sqrt 2 \times \frac{{{\mu _0}{I^2}}}{{2\pi d}} = \frac{{{\mu _0}{I^2}}}{{\sqrt 2 \pi d}}$$