Two parallel long wires carry currents \( i_1 \) and \( i_2 \) with \( i_1\gt i_2 \). When the currents are in same direction, the magnetic field at a point midway between them is \( 10\,\mu\mathrm{T} \). If the direction of \( i_2 \) is reversed, the field becomes \( 30\,\mu\mathrm{T} \). The ratio of \( i_1/i_2 \) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 : 1
The field of a long straight wire at a distance r is \( B = \mu_0 i/2\pi r \). The midway point is at the same distance from both wires, so \( B_1\propto i_1 \) and \( B_2\propto i_2 \) with the same constant. Between two parallel wires the two fields are in opposite directions when the currents are in the same direction, and in the same direction when the currents are opposite.
Currents in the same direction (fields subtract, \( i_1\gt i_2 \))
\( B_1 - B_2 = 10\,\mu\mathrm{T} \)
\( i_2 \) reversed (fields add)
\( B_1 + B_2 = 30\,\mu\mathrm{T} \)
Adding and subtracting the two equations:
\( B_1 = 20\,\mu\mathrm{T},\qquad B_2 = 10\,\mu\mathrm{T} \)
\( \displaystyle \frac{i_1}{i_2} = \frac{B_1}{B_2} = \frac{20}{10} = 2 \)