A wire having a uniform linear charge density `lambda` is bent in the form of a ring of radius `R`. Point `A` as shown in (Fig. 3.145) is in the plane of the ring but not at the center. Two elements of the ring of lengths `a_1` and `a_2` subtend very small same angle at point `A`. they are at distances `r_1` and `r_2` from point `A`, respectively `(r_2 gt r_1)`.
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Options

A. The ratio of charges of element `a_1` to that of element `a_2` is `r_1//r_s`.
B. The element `a_1` produced greater magnitude of electric field at `A` than elements `a_2`.
C. The same elements `a_1 and a_2` produce same potential at `A`
D. The direction of the net electric field produced by the elements only at `A` is toward element `a_2`.

Answer

a.,b.,c.,d.

Explanation

Charge on `a_1` is `(r_1 theta) lambda`
Charge on `a_2` is `(r_2 theta) lambda`
ratio of charged is `r_1//r_2`
`E_1 ("field produced by" a_1) = (K [(r_1 theta) lambda])/(r_1^2) = (K theta lambda)/r_(1)`
`E_2 ( "field produced by" a_2) = (K theta lambda)/r_(1)`
As`r_2 gt r_1, E_1 gt E_2` i.e., net field at `A` is toward `a_2`
`V_1 (K(r_1 theta))/r_(1) = K theta lambda`
`V_2 = (K(r_2 theta))/r_(1) = K theta lambda`
`V_1 = V_2`.