Consider a circular loop that is uniformly charged and has a radius a √2. Find the position along the positive z-axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in x y plane at the origin :

Solution

Physics sir · 4 steps

Write the axial field of a ring, set its derivative to zero, then put in the radius a√2.

1. Field on the axis. For a ring of charge Q and radius R, the sideways parts of the field from opposite bits of the ring cancel. Only the part along the axis survives, at a height z above the centre.

   E(z) = (kQ z)/(R² + z²)^(3/2)

2. Find the maximum. E = 0 at z = 0 and E → 0 far away, so there is a peak in between. Setting dE/dz = 0 gives (R² + z²) - 3z² = 0, that is R² = 2z², so z = R/√2.
3. Put in the radius. Here R = a√2, so z = a√2/√2 = a.
4. So. The field on the positive z-axis is largest at z = a, which is option (a).

The idea to keep. The axial field of a charged ring is zero at the centre and peaks at z = R/√2. Remember the formula, but always substitute the actual radius given.

Answer from the NTA final answer key. Solution by PrepWiser.