In the figure shown, the magnetic induction at the center of their arc due to the current in the potion AB will b

To find the magnetic induction at the center of an arc due to a current-carrying conductor, we can utilize the Biot-Savart Law. The scenario involves a portion of a current-carrying wire shaped as an arc.

Here's how you can approach this problem step-by-step:

1. ### Break Down the Problem

• Identify the shape of the wire and its parameters (radius, angle).
• Determine the position where you want to find the magnetic induction (center of the arc).
• Consider the current flowing through the wire.

2. ### Relevant Concepts

• The Biot-Savart Law is given by: $B = \frac{{\mu_0 I}}{4\pi} \int \frac{{d\vec{l} \times \hat{r}}}{r^2}$ Where:
• $B$ is the magnetic field.
• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \, \text{T m/A}$).
• $I$ is the current.
• $d\vec{l}$ is a small piece of the wire.
• $\hat{r}$ is the unit vector from the wire segment to the point of interest.
• $r$ is the distance from the wire segment to the point of interest.

3. ### Analysis and Detail

• For an arc of radius $R$ and angle $\theta$, the magnetic induction $B$ at the center can be simplified.
• Using the formula for a single circular arc: $B = \frac{{\mu_0 I \theta}}{4\pi R}$ Where $\theta$ is in radians.

4. ### Verify and Summarize

• Double-check that the angle $\theta$ is converted into radians if given in degrees.
• Substitute values into the magnetic induction formula for final calculation.

Final Answer

The magnetic induction $B$ at the center of the arc due to current $I$ flowing through arc $AB$ is given by: $B = \frac{{\mu_0 I \theta}}{4\pi R}$ Make sure to replace $I$, $\theta$, and $R$ with the specific values provided in the problem for an exact numerical result.