Field on axis of a uniformly charged ring of radius a with total charge Q at axial distance z. Which expression is correct?

Field on axis of a uniformly charged ring of radius a with total charge Q at axial distance z. Which expression is correct?

• A. E_z = (1/(4π ε0)) (Q z) / (z^2 + a^2)^(3/2)
• B. E_z = (1/(4π ε0)) (Q)/(z^2 + a^2)^(3/2)
• C. E_z = (1/(4π ε0)) (Q z) / (z^2 + a^2)^(1/2)
• D. E_z = (1/(4π ε0)) (Q z) / (z^2 + a^2)^(5/2)

Answer: (A)

The field on the axis comes only from the axial component of the field from each charge element, because horizontal components cancel by symmetry. For a ring element dq, the distance to the observation point is R = sqrt(z^2 + a^2), so its magnitude is dE = k dq / R^2 with k = 1/(4π ε0). The axial component is dE_z = dE cosθ, and cosθ = z / R, giving dE_z = k z dq / R^3. Since R is the same for every element around the ring, integrate over the whole ring to get E_z = k z / R^3 ∫ dq = k z Q / R^3. Substituting R^3 = (z^2 + a^2)^(3/2) yields E_z = (1/(4π ε0)) Q z / (z^2 + a^2)^(3/2). This expression makes sense physically: the field vanishes at the center (z = 0) by symmetry, and far away it behaves like a point charge along the axis, falling off as 1/z^2. The other forms miss either the projection factor or the correct power of the distance, so they don’t match the geometry of the problem.

The field on the axis comes only from the axial component of the field from each charge element, because horizontal components cancel by symmetry. For a ring element dq, the distance to the observation point is R = sqrt(z^2 + a^2), so its magnitude is dE = k dq / R^2 with k = 1/(4π ε0). The axial component is dE_z = dE cosθ, and cosθ = z / R, giving dE_z = k z dq / R^3. Since R is the same for every element around the ring, integrate over the whole ring to get E_z = k z / R^3 ∫ dq = k z Q / R^3. Substituting R^3 = (z^2 + a^2)^(3/2) yields E_z = (1/(4π ε0)) Q z / (z^2 + a^2)^(3/2). This expression makes sense physically: the field vanishes at the center (z = 0) by symmetry, and far away it behaves like a point charge along the axis, falling off as 1/z^2. The other forms miss either the projection factor or the correct power of the distance, so they don’t match the geometry of the problem.