Question

An infinitely long straight conductor carrying current 'I' is bent into a shape as shown in figure. The radius of the circular loop is 'r'. The magnetic induction at the centre of the loop at point 'O' is ______.

• A. zero
• B. $\frac{\mu_0 I}{4\pi r} (\pi - 1)$
• C. $\frac{\mu_0 I}{2\pi r} (\pi + 1)$
• D. $\frac{\mu_0 I}{2\pi r} (\pi - 1)$

Hint:

To factor an expression like $\frac{X}{2r} \pm \frac{X}{2\pi r}$, pulling out $\frac{X}{2\pi r}$ will always leave you with $(\pi \pm 1)$ inside the brackets. This is a very common algebraic trick in electromagnetism problems!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The magnetic field at the center $O$ is the sum of fields from two parts: the infinitely long straight wire and the circular loop.

Step 2: Formula Application:
Magnetic field due to a loop: $B_{loop} = \frac{\mu_0 I}{2r}$. Magnetic field due to an infinite wire: $B_{wire} = \frac{\mu_0 I}{2\pi r}$.

Step 3: Explanation:
Using the Right-Hand Thumb Rule, both fields point in the same direction at the center. $B_{net} = \frac{\mu_0 I}{2r} + \frac{\mu_0 I}{2\pi r} = \frac{\mu_0 I}{2\pi r} (\pi + 1)$.

Step 4: Final Answer:
The magnetic induction is $\frac{\mu_0 I}{2\pi r} (\pi + 1)$.