Two infinite wires carrying opposite electrical currents $I$ and $i$ are placed a distance $x$ apart. A point $P$ at a distance $y$ away from the wire carrying current $i$ is shown in the figure. If the magnetic field is zero at point $P$, then the magnitude of $i$ is

• A

  $l\left(\frac{x}{x+y}\right)$
• B

  $\prime\left(\frac{2 x}{x+y}\right)$
• Correct answer
  C

  $I\left(\frac{y}{x+y}\right)$
• D

  $\left(\frac{2 y}{x+y}\right)$

Solution

The given situation is shown in the following figure
Magnetic field at point $P$ due to the wire having current $I$ is given as $B_I=\frac{\mu_0}{2 \pi} \cdot \frac{I}{x+y}$ (downward) Magnetic field at point $P$ due to the wire having current $i$ is given as $B_i=\frac{\mu_0}{2 \pi} \cdot \frac{i}{y}$ (upward) According to question, net magnetic field at point $P$ is zero, hence $\begin{gathered} B_I=B_i \\ \Rightarrow \frac{\mu_0}{2 \pi} \cdot \frac{I}{x+y}=\frac{\mu_0}{2 \pi} \cdot \frac{i}{y} \Rightarrow \frac{I}{x+y}=\frac{i}{y} \Rightarrow i=I\left(\frac{y}{x+y}\right) \end{gathered}$

Hence, the correct answer is option: 'C'.