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Hai dây dẫn có hệ số nhiệt điện trở \({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{,}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\); ở 00C có điện trở R01, R02. Tìm hệ số nhiệt điện trở chung của hai dây khi chúng mắc:

a) Nối tiếp.

b) Song song.

Theo dõi Vi phạm
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Trả lời (1)

  • -Điện trở của hai dây dẫn ở nhiệt độ t: R1 = R01[1+\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)(t1 – t01)]; R2 = R02[1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{2}}\)(t2 – t02)].

    R1 = R01(1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t); R2 = R02(1+\[{{\text{ }\!\!\alpha\!\!\text{ }}_{2}}\]t).

    với:      t01 = t02 = 0; \({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t, \){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\)t <<1.

    -Gọi R0 là điện trở chung của hai dây dẫn ở 00C; \(\text{ }\!\!\alpha\!\!\text{ }\) là hệ số nhiệt điện trở chung của hai dây dẫn. Điện trở chung của hai dây dẫn ở nhiệt độ t là:

                R = R0(1 + \(\text{ }\!\!\alpha\!\!\text{ }\)t)                                       (1)

    a)Khi mắc nối tiếp:

                R = R1 + R2 = R01(1+\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\)t) + R02(1+\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\)t)

    =>        R = (R01 + R02) + (R01\({{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\) + R02\){{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\))t

    =>        R = (R01 + R02)\(\left[ \text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t} \right]\)              (2)

    -Từ (1) và (2) suy ra: \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).

    b)Khi mắc song song:

                R = \(\frac{{{\text{R}}_{\text{1}}}{{\text{R}}_{\text{2}}}}{{{\text{R}}_{\text{1}}}+{{\text{R}}_{\text{2}}}}=\frac{{{\text{R}}_{\text{01}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)}{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{{{\text{R}}_{\text{01}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)}+{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}\)

    =>        R = \(\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}+{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}\] = \[\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{.}\frac{\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}}{\text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}}\)

    -Với \({{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}\text{,}{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}<<\text{1}\), ta có các công thức gần đúng:

    \(\text{(1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}\text{)(1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\text{)}\approx \text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\text{; }\frac{\text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}}{\text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}}\approx \text{1}+{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{1}}}-{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{2}}}\)

    nên      \(\text{(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{t)(1}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{t)}\approx \text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{)t}\)

    =>        \(\frac{\text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{)t}}{\text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}}\approx \text{1}+\text{(}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}-\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{)t}\) \)\approx \) 1 + \)\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t}\)

    =>        R= \(\frac{{{\text{R}}_{\text{01}}}{{\text{R}}_{\text{02}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\left[ \text{1}+\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\text{t} \right]\)          (3)

    -Từ (1) và (3) suy ra: \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).

    Vậy: Hệ số nhiệt điện trở chung của hai dây khi chúng mắc nối tiếp là \(\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\); khi chúng mắc song song là \)\text{ }\!\!\alpha\!\!\text{ }=\frac{{{\text{R}}_{\text{01}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}+{{\text{R}}_{\text{02}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}{{{\text{R}}_{\text{01}}}+{{\text{R}}_{\text{02}}}}\).

      bởi Lan Anh 23/02/2022
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