Two long parallel wires carry currents of 10 A in opposite directions. They are separated by 40 cm. What is the magnetic field in the plane of the wires at a point that is 20 cm from one wire and 60 cm from the other
The Correct Answer and Explanation is :
To find the magnetic field in the plane of two long parallel wires carrying currents of 10 A in opposite directions, we can use the formula for the magnetic field due to a long straight wire:
[
B = \frac{\mu_0 I}{2 \pi r}
]
where:
- ( B ) is the magnetic field,
- ( \mu_0 ) is the permeability of free space (( 4\pi \times 10^{-7} \, \text{T m/A} )),
- ( I ) is the current in the wire (10 A in this case),
- ( r ) is the distance from the wire to the point where the magnetic field is being calculated.
Given:
- Current ( I = 10 \, \text{A} )
- Distance between the wires = 40 cm = 0.4 m
- Distance from the first wire (wire 1) to the point = 20 cm = 0.2 m
- Distance from the second wire (wire 2) to the point = 60 cm = 0.6 m
Step 1: Calculate the magnetic field due to each wire at the point of interest.
- For Wire 1 (current flowing in one direction):
[
B_1 = \frac{\mu_0 \cdot 10 \, \text{A}}{2 \pi \cdot 0.2 \, \text{m}} = \frac{4\pi \times 10^{-7} \cdot 10}{2 \pi \cdot 0.2} = \frac{2 \times 10^{-6}}{0.2} = 1 \times 10^{-5} \, \text{T}
]
The direction of this magnetic field (using the right-hand rule) will be into the page. - For Wire 2 (current flowing in the opposite direction):
[
B_2 = \frac{\mu_0 \cdot 10 \, \text{A}}{2 \pi \cdot 0.6 \, \text{m}} = \frac{4\pi \times 10^{-7} \cdot 10}{2 \pi \cdot 0.6} = \frac{2 \times 10^{-6}}{0.6} = \frac{1}{3} \times 10^{-6} \, \text{T} \approx 0.6667 \times 10^{-6} \, \text{T}
]
The direction of this magnetic field will be out of the page.
Step 2: Determine the net magnetic field at the point.
Since ( B_1 ) is directed into the page and ( B_2 ) is directed out of the page, they will oppose each other:
[
B_{\text{net}} = B_1 – B_2 = 1 \times 10^{-5} \, \text{T} – 0.6667 \times 10^{-6} \, \text{T} = 0.9333 \times 10^{-5} \, \text{T} \approx 9.33 \times 10^{-6} \, \text{T}
]
Conclusion
The net magnetic field at the point 20 cm from one wire and 60 cm from the other wire is approximately ( 9.33 \times 10^{-6} \, \text{T} ), directed into the page. This result highlights how magnetic fields interact and reinforce or cancel based on the direction of current flow in parallel conductors.