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An infinite number of charges each equa...

An infinite number of charges each equal to q, are placed along the X-axis at `x = 1, x = 2, x = 4, x = 8`,…….. and so on.
(i) find the electric field at a point `x = 0` due to this set up of charges.
(ii) What will be the electric field if the above setup, the consecutive charges have opposite signs.

Text Solution

Verified by Experts

The electric field intenistites due to position charges and due to -ve charge the fields intensity is towards the charges
` (##NAR_PHY_XII_V02_C02_S01_012_S01.png" width="80%">
The resultant intensity at the origin
`E=E_(1)+E_(3)-E_(4).......`
`E=(Q)/(4pi epsilon_(0))(1-(1)/(2^(2))+(1)/(4^(2))-(1)/(8^(2))+....)`
since the expression in the brackets is in `GP` with a common ratio`=(-1)/(2^(2))=-(1)/(4)`
`E=(Q)/(4piepsilon_(0))(1)/([1-((-1)/(4))])=(Q)/(4piepsilon_(0))(4)/(5)`
` E=(4)/(5)(Q)/(4piepsilon_(0))`
`E=(Q)/(5piepsilon_(0))`
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