A circuit connected to an ac source of emf e = e _ 0 sin (100t)...

$\omega = 100$ rad/sec

phase difference between $e and i$ is given by

$tan\theta = \dfrac{X}{R} ∴ \theta = \dfrac{\pi}{4}$

so $tan\theta = tan\dfrac{\pi}{4} = 1$

in case of $RL$ circuit $X = {X}_{L} = \omegaL$

And in case of $RC$ circuit $X = {X}_{c} = \dfrac{1}{\omegac}$

For option \left(A\right): R = 1000Ω$and</p><p>${X}_{C} = \dfrac{1}{\omegaC} = \dfrac{1}{\left(100\right)\left(1 \times {10}^{- 6}\right)} = 10000Ω$<br />$tan \theta = \dfrac{{X}_{C}}{R} \ne 1$<br /><strong>For option</strong> \left(B\right): R = {10}^{3}Ω$ and

${X}_{L} = L\omega = \left({10}^{- 3}\right)\left(100\right) = 0.1Ω$

$tan \theta = \dfrac{{X}_{L}}{R} \ne 1$
For option \left(C\right): R = 1000Ω$and</p><p>${X}_{L} = \omegaL = \left(100\right)\left(10\right) = 1000Ω$<br />$tan \theta = \dfrac{{X}_{L}}{R} \ne 1$<br /><strong>For option</strong>\left(D\right): R = 1kΩ = 1000Ω$ and

${X}_{C} = \dfrac{1}{\omegaC} = \dfrac{1}{\left(100\right)\left(10 \times {10}^{- 6}\right)} = 1000Ω$
$tan \theta = \dfrac{{X}_{C}}{R} = 1; \theta = {45}^{o}$