For a uniformly charged ring of radius a with total charge Q, when z = 0 along the axis, what is E_z?

For a uniformly charged ring of radius a with total charge Q, when z = 0 along the axis, what is E_z?

• A. E_z = 0
• B. E_z = (1/(4π ε0)) (Q z) / (z^2 + a^2)^(3/2)
• C. E_z = (1/(4π ε0)) (Q)/(z^2 + a^2)^(3/2)
• D. E_z = (1/(4π ε0)) (Q z) / (z^2 + a^2)^(1/2)

Answer: (A)

Symmetry of a circular charge distribution and how fields add along the axis. At the center of the ring, every charge element lies in the plane of the ring, and the field from each element points toward the center in that plane. The projection of that field onto the axis (the z-direction) is zero for every element, so all contributions cancel and the net axial field is zero. Equivalently, the standard axial-field expression is E_z(z) = (1/(4π ε0)) Q z / (z^2 + a^2)^{3/2}. Plugging in z = 0 gives E_z = 0, which matches the symmetry argument.

Symmetry of a circular charge distribution and how fields add along the axis. At the center of the ring, every charge element lies in the plane of the ring, and the field from each element points toward the center in that plane. The projection of that field onto the axis (the z-direction) is zero for every element, so all contributions cancel and the net axial field is zero.

Equivalently, the standard axial-field expression is E_z(z) = (1/(4π ε0)) Q z / (z^2 + a^2)^{3/2}. Plugging in z = 0 gives E_z = 0, which matches the symmetry argument.