B

Step 1: Understanding the problem involves calculating the electric field due to a uniformly charged thread. We assume the tension in the thread leads to an electric field at point O due to the segment of the thread within a radius r.

Step 2: The electric field due to a uniformly charged line is given by the formula:
$$ E = \frac{\lambda}{2\pi\epsilon_0 r} $$
where λ is the linear charge density and ε₀ is the permittivity of free space.

Step 3: Considering the direction of the electric field at point O, we can observe that the electric field produced by each infinitesimal segment of the thread contributes to the net electric field. The components in the vertical direction will cancel out due to symmetry, while the horizontal components will add up.

Step 4: By evaluating this for one configuration (for instance, configuration (a)), we see a net effect on the electric field pointing towards the center due to the thread curvature. Similarly, configuration (b) also contributes to the electric field at a similar value at point O.

Step 5: Since these two configurations are symmetric and yield the same magnitude of electric field produced at point O, we conclude:
Therefore, the final expression for the magnitude of the electric field strength at point O equals that from a uniformly charged line, confirming:
$$ E = \frac{\lambda}{2\pi\epsilon_0 r} $$
is represented adequately in option B.