Question

Two long parallel straight metal wires \(A\) and \(B\) carrying currents \(12\text{ A}\) and \(36\text{ A}\) respectively, in the same direction are separated by \(50\text{ cm}\). The point relative to \(A\), where the resultant magnetic induction between the two wires due to the currents is zero, will be

• A. \(90\text{ cm}\)
• B. \(7.5\text{ cm}\)
• C. \(28\text{ cm}\)
• D. \(12.5\text{ cm}\)

Hint:

For two parallel wires carrying currents in the same direction, the neutral point lies between the wires and closer to the wire carrying smaller current.

Answer 1

The Correct Option is D

Step 1: Set up the problem.
We have two long parallel wires A and B carrying currents $ I_A = 12\text{ A} $ and $ I_B = 36\text{ A} $ in the same direction, separated by $ d = 50\text{ cm} $. We want the point P between them where the net magnetic field is zero. Since both currents flow in the same direction, fields due to each wire at a point between them point in opposite directions (by the right-hand rule), so cancellation is possible between the wires.
Step 2: Assign a variable for position.
Let P be at distance $ x $ cm from wire A, so it is $ (50 - x) $ cm from wire B. We restrict $ 0 < x < 50 $ since cancellation only happens between the wires for parallel currents.
Step 3: Write the magnetic field due to each wire at P.
The magnetic field due to a long straight wire at distance $ r $ is \[ B = \frac{\mu_0 I}{2\pi r} \] So the field magnitudes at P are \[ B_A = \frac{\mu_0 \cdot 12}{2\pi x}, \qquad B_B = \frac{\mu_0 \cdot 36}{2\pi(50-x)} \]
Step 4: Apply the zero-field condition.
For the fields to cancel, their magnitudes must be equal: \[ B_A = B_B \implies \frac{12}{x} = \frac{36}{50 - x} \] Cross-multiplying: \[ 12(50 - x) = 36x \implies 600 = 48x \implies x = 12.5\text{ cm} \]
Step 5: Verify the result makes sense.
Wire A carries less current (12 A), so the zero-field point is closer to A, at $ x = 12.5\text{ cm} $, which is much closer to A than to B. This is physically consistent because a weaker current source requires a closer observation point to match the field of the stronger source.
Step 6: State the answer.
The point where the resultant magnetic field is zero is at $ 12.5\text{ cm} $ from wire A. \[ \boxed{12.5\text{ cm from wire A}} \]