Find Electric Field along z-axis

Question

An infinite number of identical charged rings of radius w and charge Q are arranged coaxially along the z- axis such that each adjacent pair is a distance w apart and one lies with the origin in its plane. Find a series expression for the electric field at all points along the z-axis.

Answer 1

Hi Boni

 

Electric field of a charged ring is given as ( For the one at origin field will be zero Ez₀ =0)

 

Ez= KQz/ ( w² +z ²) ^3/2 

 

Case 1 : when ring is not very small in radius w= radius of ring and z = distance of center ring from the origin

 

Ez₁₌ KQw/(w² +w ²) ^3/2

 

Ez₂ = KQ(2w)/(w² +4 w ²) ^3/2

 

Ez₃ = KQ(3w) / (w² +9w ²) ^3/2

 

and so on.... add this sequence

 

Case 2 For small radius

 

Ez₁ = KQ/w ²

Ez₂ = KQ/4w ²

Ez₃ = KQ/9w ²

 

and so on....

Add this sequence

 

Case 3 ( Applicable to both)

When the charged rings are place on both sides ( +z and -z)

The net Electric field at the origin will be zero.