Question

Two very long straight conductors (wires) are set parallel to each other. Each carries a current ' I ' in the same direction and the separation between them is ' 2 r '. The intensity of the magnetic field at point ' P ' (as shown in the figure) ( \(\mu_0=\) permeability of free space) is

• A. \(\frac{2}{3}\frac{\mu _{0}I}{\pi r}\)
• B. \(\frac{3}{8}\frac{\mu _{0}I}{\pi r}\)
• C. \(\frac{1}{4}\frac{\mu _{0}\mathrm{I}}{\pi \mathrm{r}}\)
• D. \(\frac{\mu _{0}I}{2\pi r}\)

Hint:

Add the fields of the two wires at P; both point the same way, using distances r and 3r.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A long straight wire gives \(B=\dfrac{\mu_0 I}{2\pi d}\) at distance \(d\).
P is at distance \(r\) from the near wire and \(r+2r=3r\) from the far wire.

Step 2: Direction
Both currents flow upward and P lies to the right of both wires, so both fields point the same way and add.

Step 3: Calculation
\(B=\dfrac{\mu_0 I}{2\pi r}+\dfrac{\mu_0 I}{2\pi (3r)}=\dfrac{\mu_0 I}{2\pi r}\left(1+\dfrac{1}{3}\right)=\dfrac{2}{3}\dfrac{\mu_0 I}{\pi r}\)

Final Answer:
The magnetic field at P is two thirds of mu naught I over pi r. \[ \boxed{\text{(A) }\dfrac{2}{3}\dfrac{\mu_{0}I}{\pi r}} \]