Question

Two long straight parallel wires A and B separated by \( 5 \) m carry currents \( 2 \) A and \( 6 \) A respectively in the same direction. The resultant magnetic field due to the two wires at a point \( 2 \) m distance from the wire A in between the two wires is?

• A. \( 2 \times 10^{-6} \) T
• B. \( 2 \times 10^{-7} \) T
• C. \( 4 \times 10^{-7} \) T
• D. \( 4 \times 10^{-6} \) T \

Hint:

For two parallel current-carrying wires, the magnetic field at a point between them is found by considering the direction of the fields using the right-hand rule and subtracting when they oppose each other.

Answer 1

The Correct Option is B

Step 1: Magnetic field due to a long current-carrying wire
The magnetic field at a distance \( r \) from a long, straight current-carrying wire is given by: \[ B = \frac{\mu_0 I}{2 \pi r} \] where: - \( \mu_0 = 4\pi \times 10^{-7} \) Tm/A is the permeability of free space, - \( I \) is the current in the wire, - \( r \) is the perpendicular distance from the wire. Step 2: Compute the magnetic fields due to both wires
Let wire A carry \( I_A = 2 \) A and wire B carry \( I_B = 6 \) A. The separation between the wires is \( 5 \) m. The point of interest is \( 2 \) m from wire A. The magnetic field due to wire A at this point is: \[ B_A = \frac{(4\pi \times 10^{-7}) \times 2}{2\pi \times 2} \] \[ = \frac{8\pi \times 10^{-7}}{4\pi} \] \[ = 2 \times 10^{-7} \text{ T} \] The distance of the point from wire B is: \[ r_B = 5 - 2 = 3 \text{ m} \] The magnetic field due to wire B at this point is: \[ B_B = \frac{(4\pi \times 10^{-7}) \times 6}{2\pi \times 3} \] \[ = \frac{24\pi \times 10^{-7}}{6\pi} \] \[ = 4 \times 10^{-7} \text{ T} \] Step 3: Determine the net magnetic field
Since both currents are in the same direction, their magnetic fields at the given point will oppose each other (by the right-hand rule). The resultant magnetic field is: \[ B_{\text{net}} = B_B - B_A \] \[ = (4 \times 10^{-7} - 2 \times 10^{-7}) \text{ T} \] \[ = 2 \times 10^{-7} \text{ T} \] Thus, the resultant magnetic field at the given point is \( 2 \times 10^{-7} \) T. \bigskip