Force Calculation for Uniform Ring Charge

The question asks for the magnitude of the force on a point charge $q$ placed at the center of a uniform ring carrying a total charge $Q$ and having radius $R$. We need to consider the electric field produced by the ring at its center.

Electric Field at the Center due to Symmetry

A uniform ring charge possesses a high degree of symmetry. Consider any small charge element $dq$ located at a point P on the ring. The electric field $d\vec{E}$ produced by this element at the center O has a certain magnitude and direction.

Now, consider the charge element $dq'$ located diametrically opposite to $dq$ on the ring. The electric field $d\vec{E'}$ produced by $dq'$ at the center O will have the same magnitude as $d\vec{E}$ but will point in the exact opposite direction.

Due to this symmetry:

• For every charge element $dq$, there is a corresponding diametrically opposite element $dq'$.
• The electric fields produced by these pairs of elements at the center cancel each other out ($d\vec{E} + d\vec{E'} = 0$).

Therefore, the net electric field $\vec{E}$ at the center of the uniform ring is zero.

Force on the Point Charge

The force $\vec{F}$ experienced by a point charge $q$ in an electric field $\vec{E}$ is given by:

$ \vec{F} = q\vec{E} $

Since the electric field $\vec{E}$ at the center of the ring is zero:

$ \vec{F} = q \times 0 = 0 $

Thus, the magnitude of the force on the point charge $q$ at the center is $0$.

Conclusion

Based on the principle of symmetry, the net electric field at the center of a uniform charged ring is zero. Consequently, the force exerted on a point charge placed at the center is also zero.