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Two long parallel wires carrying currents 2.5 A and I (ampere) in the same direction(directed into the plane of the paper) are held at P and Q respectively such that they are perpendicular to the plane paper. The points P and Q are located at a distance of 5m and 2m respectively from a collinear point R(see figure) .
Given image
Find all the positions at which a third long parallel wire carrying a current of magnitude $2.5 A$ may be placed, so that the magnetic induction at $R$ is zero.

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$r = 1 m$ where it can be placed at $M$ and $N$

$r = 2 m$ where it can be placed at $M$ and $N$

$r = 0.1 m$ where it can be placed at $M$ and $N$

$r = 12 m$ where it can be placed at $M$ and $N$

answer is A.

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Detailed Solution

Net field at $R$ due to wires $P$ and $Q$ is

$B = 10^{- 7} (I + 1) T = 5 \times 10^{- 7} T$

Magnetic field due to third wire carrying a current of $2.5 A$ should be $5 \times 10^{- 7} T$ in upward direction so that net field at $R$ becomes zero. Let distance of this wire from $R$ be $r$ . Then,

$\frac{μ_{0}}{2 π} \frac{2.5}{r} = 5 \times 10^{- 7} \frac{(2 \times 10^{- 7}) (2.5)}{r} = 5 \times 10^{- 7} r = 1 m$

So, the third wire can be put at $M$ or $N$ as shown in the figure.

If it is placed at $M$ , then current in it should be outwards and if placed at $N$ , then current is inwards.

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