In the figure, currents are flowing in opposite directions in two parallel conductors. What should be the current \( I \) in the conductor X, so that the resultant magnetic field at the point P is zero? Current in the conductor Y is 10 ampere.
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The magnetic fields from two current-carrying conductors in opposite directions can cancel each other out if the magnitudes of the fields are equal and they are opposite in direction.
Step 1: Magnetic Field Due to a Current-Carrying Conductor.
The magnetic field \( B \) at a point due to a current \( I \) in a long straight conductor is given by Ampere's law:
\[
B = \frac{\mu_0 I}{2 \pi r}
\]
where:
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current,
- \( r \) is the distance from the wire to the point where the field is being measured.
Step 2: Magnetic Field at Point P Due to Conductor Y.
The magnetic field at point P due to conductor Y, with current \( 10 \, \text{A} \) and distance \( 5 \, \text{cm} = 0.05 \, \text{m} \), is:
\[
B_Y = \frac{\mu_0 \times 10}{2 \pi \times 0.05}
\]
\[
B_Y = \frac{4 \times 10^{-7} \times 10}{2 \pi \times 0.05} = \frac{4 \times 10^{-6}}{0.314} = 1.27 \times 10^{-5} \, \text{T}
\]
Step 3: Magnetic Field at Point P Due to Conductor X.
For the field to be zero at point P, the magnetic field due to conductor X must be equal and opposite in direction to the field due to conductor Y. Let the current in conductor X be \( I \), and the distance from point P to conductor X is \( 10 \, \text{cm} = 0.1 \, \text{m} \). The magnetic field at point P due to conductor X is:
\[
B_X = \frac{\mu_0 I}{2 \pi \times 0.1}
\]
Step 4: Setting the Fields Equal.
For the magnetic fields to cancel each other out, we set \( B_X = B_Y \):
\[
\frac{\mu_0 I}{2 \pi \times 0.1} = 1.27 \times 10^{-5}
\]
Solving for \( I \):
\[
I = \frac{1.27 \times 10^{-5} \times 2 \pi \times 0.1}{\mu_0} = \frac{1.27 \times 10^{-5} \times 0.2 \pi}{4 \times 10^{-7}} = \frac{1.27 \times 10^{-5} \times 0.628}{4 \times 10^{-7}}
\]
\[
I = 20 \, \text{A}
\]
Step 5: Conclusion.
Thus, the current in conductor X must be \( 20 \, \text{A} \) in the opposite direction to cancel the magnetic field at point P.
