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

To provide a detailed solution to the problem regarding magnetic induction at the center of an arc due to current, I will proceed with the steps outlined.

1. Break Down the Problem

• Identify the geometry of the problem, including the relevant arc and center.
• Recognize the current through the wire and its contribution to magnetic induction.

2. Relevant Concepts

The magnetic induction (magnetic field) $B$ at the center of a circular arc carrying current $I$ can be calculated using the formula: $B = \frac{\mu_0 I \theta}{4\pi r}$ where:

• $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \, \text{T m/A}$)
• $\theta$ is the angle in radians subtended by the arc
• $r$ is the radius of the arc.

3. Analysis and Detail

Assuming we have:

• A current $I$ along the arc $AB$.
• The length of the arc $AB$ subtends an angle $\theta$ at the center.
• The radius of the arc is $r$.

Substituting the given values into the formula, we will perform our calculations based on the input parameters (current value $I$, angle $\theta$, and radius $r$).

4. Verify and Summarize

Once we substitute the numbers, it is imperative to check each value carefully to ensure the magnetic induction calculation is accurate.

Final Answer

After substituting the values into the formula and simplifying, the magnetic induction at the center due to the current $I$ in the portion $AB$ will yield a numerical result for $B$.

To provide the exact result, please provide the values of current $I$, angle $\theta$, and radius $r$.