An electric field of magnitude E is produced at the center of a uniformly charged ring by one half of the ring. What is the magnitude of the total electric field at the center of the ring?

Step 1: Analyze Question 1.
The electric field at the center of a uniformly charged ring is zero. This means the vector sum of the electric fields due to all parts of the ring at the center is zero.
Let $\vec{E}_{AKB}$ be the electric field at $O$ due to the part $AKB$, and $\vec{E}_{ACDB}$ be the electric field at $O$ due to the part $ACDB$.
We are given that $\vec{E}_{AKB}$ has magnitude $E$ and is directed along $KO$.
The total electric field at $O$ is $\vec{E}_{total} = \vec{E}_{AKB} + \vec{E}_{ACDB} = 0$.
Therefore,