Câu hỏi 45

Đáp án C

Ta có :\(\omega _0^2 = {\omega _L}.{\omega _C};{\omega _C} = \sqrt {{1 \over {LC}} - {{{R^2}} \over {2{L^2}}}} \)

Ta có:

\(\left\{ \matrix{
{\omega _1} = 2\pi {f_1} = 2\pi {n_1}p = 40\pi (rad/s) \hfill \cr
{\omega _2} = 2\pi {f_2} = 2\pi {n_2}p = 120\pi (rad/s) \hfill \cr} \right.\)

Điện áp hiệu dụng hai đầu cuộn cảm :\({U_L} = I.{Z_L} = {{E.{Z_L}} \over {\sqrt {{R^2} + {{\left( {{Z_L} - {Z_C}} \right)}^2}} }} = {{NBS{\omega ^2}L} \over {\sqrt 2 \sqrt {{R^2} + {{\left( {{Z_L} - {Z_C}} \right)}^2}} }}\)

Khi

\(\eqalign{
& {U_{{L_1}}} = {U_{{L_2}}} = > {{\omega _1^2} \over {\sqrt {{R^2} + {{\left( {{\omega _1}L - {1 \over {{\omega _1}C}}} \right)}^2}} }} = {{\omega _2^2} \over {\sqrt {{R^2} + {{\left( {{\omega _2}L - {1 \over {{\omega _2}C}}} \right)}^2}} }} \cr
& \Leftrightarrow {R^2} + {\left( {{\omega _2}L - {1 \over {{\omega _2}C}}} \right)^2} = 81{R^2} + 81{\left( {{\omega _1}L - {1 \over {{\omega _1}C}}} \right)^2} \cr
& \omega _2^2{L^2} - {{2L} \over C} + {1 \over {\omega _2^2{C^2}}} = 80{R^2} + 81\omega _1^2{L^2} - {{162L} \over C} + {{81} \over {\omega _1^2{C^2}}} \cr
& 160{L \over C} - 80{R^2} = \left( {81\omega _1^2 - \omega _2^2} \right){L^2} + {1 \over {{C^2}}}\left( {{{81} \over {\omega _1^2}} - {1 \over {\omega _2^2}}} \right) \cr
& 160{1 \over {LC}} - 80{{{R^2}} \over {{L^2}}} = \left( {81\omega _1^2 - \omega _2^2} \right) + {1 \over {{L^2}{C^2}}}\left( {{{81} \over {\omega _1^2}} - {1 \over {\omega _2^2}}} \right) \cr
& 160\left( {{1 \over {LC}} - {{{R^2}} \over {2{L^2}}}} \right) = \left( {81\omega _1^2 - \omega _2^2} \right) + {1 \over {{L^2}{C^2}}}\left( {{{81} \over {\omega _1^2}} - {1 \over {\omega _2^2}}} \right) \cr} \)

Lại có 

\(\left\{ \matrix{
\omega _0^2 = {1 \over {LC}} \hfill \cr
\omega _C^2 = {1 \over {LC}} - {{{R^2}} \over {2{L^2}}} \hfill \cr} \right. = > 160\omega _C^2 = \left( {81\omega _1^2 - \omega _2^2} \right) + \omega _0^4\left( {{{81} \over {\omega _1^2}} - {1 \over {\omega _2^2}}} \right)\left( * \right)\)

Thay \(\omega _0^2 = {\omega _L}.{\omega _C}\) vào (*) ta có : \(160{\left( {{{\omega _0^2} \over {{\omega _L}}}} \right)^2} = \left( {81\omega _1^2 - \omega _2^2} \right) + \omega _0^4\left( {{{81} \over {\omega _1^2}} - {1 \over {\omega _2^2}}} \right)\)

Thay số ta có :\(160{\left( {{{\omega _0^2} \over {48\pi }}} \right)^2} = \left( {81.{{\left( {40\pi } \right)}^2} - {{\left( {120\pi } \right)}^2}} \right) + \omega _0^4\left( {{{81} \over {{{\left( {40\pi } \right)}^2}}} - {1 \over {{{\left( {120\pi } \right)}^2}}}} \right) =  > {\omega _0} \approx 156,12rad/s\)