Question

In the following figure magnitude of the magnetic field at the point p is ______.

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

Hint:

Break complex wire shapes into simple segments: straight lines, arcs, and circles. Any straight wire segment that aligns perfectly so it points directly at (or away from) the point of interest contributes exactly zero magnetic field!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The total magnetic field at point $P$ is the vector sum of fields due to the straight wire and the circular arc.

Step 2: Formula Application:
For a semi-infinite wire, $B_{wire} = \frac{\mu_0 I}{4\pi r}$.
For a quarter-circle arc (angle $\theta = \pi/2$), $B_{arc} = \frac{\mu_0 I}{4\pi r} \times \theta = \frac{\mu_0 I}{4\pi r} \times \frac{\pi}{2} = \frac{\mu_0 I}{8r}$. (Commonly represented as $1/4$ of a full circle: $\frac{1}{4} \frac{\mu_0 I}{2r}$).

Step 3: Explanation:
Assuming both fields are in the same direction (into the page), $B_{net} = \frac{\mu_0 I}{4\pi r} + \frac{\mu_0 I}{8r}$. If the arc is a semi-circle, the second term is $\frac{\mu_0 I}{4r}$. Based on the options provided, the geometry corresponds to a semi-infinite wire and a semi-circular arc component.

Step 4: Final Answer:
The magnitude is $\frac{\mu_0 I}{4\pi r} + \frac{\mu_0 I}{4r}$.