Cho đoạn mạch mắc nối tiếp gồm biến trở R, cuộn cả...

A

- Hướng dẫn giải

Phương pháp giải:

Công thức tính công suất  

\(P = U.I.\cos \varphi \)

Giải chi tiết:

Công suất khi R = R₁ là:  

\({P_1} = U.{I_1}.\cos {\varphi _1} = \frac{{{U^2}}}{{\sqrt {{R_1}^2 + {{\left( {{Z_L} - {Z_C}} \right)}^2}} }}.\cos {\varphi _1}\)

Công suất khi R = R₂ là: 

\({P_2} = U.{I_2}.\cos {\varphi _2} = \frac{{{U^2}}}{{\sqrt {{R_2}^2 + {{\left( {{Z_L} - {Z_C}} \right)}^2}} }}.\cos {\varphi _2}\)

Từ đề bài ta có:

\(\begin{array}{l}
{\cos ^2}{\varphi _1} + {\cos ^2}{\varphi _2} = \frac{7}{{10}}\\
\Rightarrow \frac{1}{{1 + {{\tan }^2}{\varphi _1}}} + \frac{1}{{1 + {{\tan }^2}{\varphi _2}}} = \frac{7}{{10}} \Leftrightarrow \frac{1}{{1 + {{\left( {\frac{{{Z_{LC}}}}{{{R_1}}}} \right)}^2}}} + \frac{1}{{1 + {{\left( {\frac{{{Z_{LC}}}}{{{R_2}}}} \right)}^2}}} = \frac{7}{{10}}\\
\Leftrightarrow \frac{1}{{1 + {{\left( {\frac{{{Z_{LC}}}}{{{R_1}}}} \right)}^2}}} + \frac{1}{{1 + {{\left( {\frac{{{Z_{LC}}}}{{\frac{1}{2}{R_1}}}} \right)}^2}}} = \frac{7}{{10}} \Leftrightarrow \frac{1}{{1 + {{\left( {\frac{{{Z_{LC}}}}{{{R_1}}}} \right)}^2}}} + \frac{1}{{1 + 4{{\left( {\frac{{{Z_{LC}}}}{{{R_1}}}} \right)}^2}}} = \frac{7}{{10}}
\end{array}\)

Đặt  \({\left( {\frac{{{Z_{LC}}}}{{{R_1}}}} \right)^2} = x;\,\,\left( {x > 0} \right)\) ta được:

\(\begin{array}{l}
\frac{1}{{1 + x}} + \frac{1}{{1 + 4x}} = \frac{7}{{10}} \Rightarrow \frac{{5x + 2}}{{4{x^2} + 5x + 1}} = \frac{7}{{10}}\\
\Rightarrow 10.(5x + 2) = 7.(4{x^2} + 5x + 1)\\
\Leftrightarrow 28{x^2} - 15x - 13 = 0 \Rightarrow \left[ \begin{array}{l}
{x_1} = 1\,\,\,(tm)\\
{x_2} = - \frac{{13}}{{28}}\,\,(loai)
\end{array} \right.
\end{array}\)

Vậy:

\(\begin{array}{l}
\frac{{{Z_{LC}}}}{{{R_1}}} = 1 \Leftrightarrow {Z_{LC}} = {R_1} \Rightarrow \tan {\varphi _1} = 1 \Rightarrow {\varphi _1} = {45^0} \Rightarrow \cos {\varphi _1} = \frac{1}{{\sqrt 2 }}\\
\tan {\varphi _2} = 2\tan {\varphi _1} = 2 \Rightarrow \cos {\varphi _2} = \sqrt {\frac{1}{{1 + \tan {\varphi _2}}}} = \frac{1}{{\sqrt 5 }}
\end{array}\)

Tỉ số :

\(\begin{array}{l}
\frac{{{P_1}}}{{{P_2}}} = \frac{{{U^2}.\cos {\varphi _1}}}{{\sqrt {{R_1}^2 + Z_{LC}^2} }}.\frac{{\sqrt {R_2^2 + Z_{LC}^2} }}{{{U^2}.\cos {\varphi _2}}} = \frac{1}{{\sqrt 2 }}.\frac{1}{{\sqrt {{R_1}^2 + R_1^2} }}.\frac{{\sqrt {\frac{1}{4}R_1^2 + R_1^2} }}{{\frac{1}{{\sqrt 5 }}}} = \frac{5}{4}\\
\end{array}\)

Chọn A.