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Question

# Two parallel long wires carry currents i1 and i2 with i1>i2. when the currents are in the same direction, the magnetic field at a point midway between the wires is 10 μT. If the direction of i2 is reversed, the field becomes 30 μT. The ratio i1i2 is

A
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B
3
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C
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D
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Solution

## The correct option is C 2Initially when wires carry currents in the same direction, the magnetic field at mid point O due to wire 1 and 2 respectively are, B1=μ0i12πx (Inwards to the plane of two wires) B2=μ0i22πx (Outward to the plane of wire) ∵i1>i2 hence B1>B2 and the direction of B1 & B2 are opposite ⇒Bnet=B1−B2 or, Bnet=μ02πx(i1−i2) Also, given in question; μ02πx(i1−i2)=10×10−6 ............ (1) If the direction of i2 is reversed then, B1=μ0i2πx B2=μ0i22πx Both the fields are perpendicularly inward to the plane of wires. ⇒Bnet=B1+B2 or, Bnet=μ02πx(i1+i2) According to question, in this case Bnet=30×10−6 T μ0(i1+i2)2πx=30×10−6....(2) Dividing (2) by (1) i1+i2i1−i2=30×10−610×10−6 or, i1+i2i1−i2=31 or, 3i1−3i2=i1+i2 or, 2i1=4i2 ∴i1i2=21 Why this Question ? Note: The magnetic field due to a long current carrying wire at a perpendicular distance x is given by, B=μ0i2πx In order to find the net field due to two or more wires, apply superposition principle by carefully taking into account, the direction of magnetic field due to different wires

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