Question

An infinite number of charges , each equal to q are placed along the a-axis at x=1 , 2 , 4 , 64 ---- and so on
1) if all the charges are of same sign , what is 'E' at orgin.?
if Ist charge is +ve , IInd charge is negative, third charge is positive --- so on .What is the E at orgin ?

Answer 1

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Answer 2

(a) \[\[\[E = q/(4\pi \epsilon _{0})\sum_{n = 0}^{\infty}(1/4^{n})\]\]\]
The intinite sum is a geometric series and adds up to 1/(1 - 1/4) = 4/3. Thus
\[E = (q/(4\pi \epsilon _{0})))(4/3) = q/(3\pi \epsilon _{0})\]
The electric field intensity at x = 0 is \[q/(3\pi \epsilon _{o}) N * m ^{2}/C\]

(b) If the even-munbered charges all toggle to -q, then the geometric series altermates. The only thing that changes here is, shereas the denominator in the first sum was 1 - 1/4, here it is 1 + 1/4, so 1/(1 + 1/4) = 4/5.
Therefore, the electric field strength at x = 0 is \[q/(5\pi \epsilon _{o})\]