The `A.C` circuit shown in figure. Find the frequency `(w_(0))` of the `AC` voltage source so that current through the source will be in same phase with the voltage of source.

Options

A. `W_(0) = sqrt((1)/(LC) + (R^(2))/(L^(2)))`
B. `W_(0) = sqrt((1)/(LC) - (R^(2))/(L^(2)))`
C. `W_(0) = (1)/(sqrt(LC))`
D. `W_(0) = sqrt((1)/(LC) + (R^(2))/(4L) + (R )/(2L))`

Answer

B

Explanation

If `i_(1)` and `i_(2)` are current in upper and lower branch than
if `vec(i_(1))` and `vec(i_(2))` have to be in phase with `vec(V)` then `i_(1) sin phi = i_(2)`
`(V_(0))/(sqrt(R^(2) + X_(L)^(2))) (X_(L))/(sqrt(R^(2) + X_(L)^(2))) = (V_(0))/(X_(C ))`
`V_(0) = sqrt((1)/(LC) - ((R )/(L))^(2)) (b)`