Two parallel long current carrying wires separated by a distance $2r$ are shown in the figure. The ratio of the magnetic field at $A$ to the magnetic field produced at $C$ is $\dfrac{x}{7}$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 5
Both currents are in the same direction, so at points outside the pair (A and C) the two fields add.
At $A$: distance $r$ from $I$ and $3r$ from $2I$:
$$B_A = \frac{\mu_0}{2\pi}\left(\frac{I}{r} + \frac{2I}{3r}\right) = \frac{\mu_0I}{2\pi r}\cdot\frac53$$
At $C$: distance $3r$ from $I$ and $r$ from $2I$:
$$B_C = \frac{\mu_0}{2\pi}\left(\frac{I}{3r} + \frac{2I}{r}\right) = \frac{\mu_0I}{2\pi r}\cdot\frac73$$
$$\frac{B_A}{B_C} = \frac57 \Rightarrow x = 5$$
Solution by Sreeraj P, M.Sc Physics