If a direct current of value a ampere is superimposed on an alternative current I=b `sin omega t` flowing through a wire, what is the effective value of the resulting current in the circuit?

Options

A. `[a^(2)-1/2 b^(2)]^(1//2)`
B. `[a^(2)+b^(2)]^(1//2)`
C. `[(a^(2))/(2)+b^(2)]^(1//2)`
D. `[a^(2)+1/2 b^(2)]^(1//2)`

Answer

D

Explanation

As current at any instant in the circuit will be
`I=(I_(dc))+(I_(ac))=a+b sin omega t`
so,` I_(eff) =[(int_(0)^(T) I^(2)dt)/(int_(0)^(T)dt)]^(1//2)=[(1)/(T) int_(0)^(2) (a+b sin omega t)^(2)dt]^(1//2)`
i.e.,` I_(eff)=[(1)/(T) int_(0)^(2) (a^(2)+2ab sin omega t+b^(2)sin^(2) omega t)dt]^(1//2)`
but as `(1)/(T) int _(0)^(T) sin omega t dt = 0 and 1/T int_(0)^(T)sin^(2) omega t dt = 1/2 `
so, `I_(eff) = [a^(2) +1/2 b^(2)]^(1//2)`.