This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Ili kupata mikondo I_1 na I_2, tutatumia Sheria za Kirchhoff (Kirchhoff's Laws), hasa Sheria ya Voltage ya Kirchhoff (KVL) kwa kila kitanzi (mesh). Step 1: Fafanua mikondo ya kitanzi (mesh currents) na tumia KVL. Tutaweka mikondo mitatu ya kitanzi, i_1, i_2, na i_3, zote zikizunguka kwa mwelekeo wa saa (clockwise) kama inavyoonyeshwa kwenye mchoro. KVL kwa Kitanzi cha Kushoto (Loop 1): Kuanzia kona ya chini kushoto na kusonga kwa mwelekeo wa saa: -10V + (3)i_1 + (5)(i_1 - i_2) = 0 3i_1 + 5i_1 - 5i_2 = 10 8i_1 - 5i_2 = 10 (Equation 1) KVL kwa Kitanzi cha Kati (Loop 2): Kuanzia katikati chini na kusonga kwa mwelekeo wa saa: -(5)(i_1 - i_2) - 15V + (9)(i_2 - i_3) = 0 -5i_1 + 5i_2 - 15 + 9i_2 - 9i_3 = 0 -5i_1 + 14i_2 - 9i_3 = 15 (Equation 2) KVL kwa Kitanzi cha Kulia (Loop 3): Kuanzia kona ya chini kulia na kusonga kwa mwelekeo wa saa: -(9)(i_2 - i_3) + (4)i_3 + 25V = 0 -9i_2 + 9i_3 + 4i_3 = -25 -9i_2 + 13i_3 = -25 (Equation 3) Step 2: Tatua mfumo wa milinganyo ya linear. Tuna mfumo wa milinganyo mitatu: 1) 8i_1 - 5i_2 = 10 2) -5i_1 + 14i_2 - 9i_3 = 15 3) -9i_2 + 13i_3 = -25 Kutoka Mlinganyo (1), tunaweza kutoa i_1: i_1 = (10 + 5i_2)/(8) Kutoka Mlinganyo (3), tunaweza kutoa i_3: i_3 = (-25 + 9i_2)/(13) Badilisha i_1 na i_3 kwenye Mlinganyo (2): -5((10 + 5i_2)/(8)) + 14i_2 - 9((-25 + 9i_2)/(13)) = 15 Ili kuondoa madhehebu, zidisha mlinganyo mzima kwa 8 × 13 = 104: -5 × 13 (10 + 5i_2) + 14 × 104 i_2 - 9 × 8 (-25 + 9i_2) = 15 × 104 -65(10 + 5i_2) + 1456i_2 - 72(-25 + 9i_2) = 1560 -650 - 325i_2 + 1456i_2 + 1800 - 648i_2 = 1560 Kusanya vigezo vya i_2 na namba za kawaida: (1456 - 325 - 648)i_2 + (1800 - 650) = 1560 483i_2 + 1150 = 1560 483i_2 = 1560 - 1150 483i_2 = 410 i_2 = (410)/(483) ≈ 0.84886 A Sasa tafuta i_1: i_1 = (10 + 5i_2)/(8) = (10 + 5(410/483))/(8) = (10 + 2050/483)/(8) = ((4830 + 2050)/483)/(8) = (6880)/(3864) ≈ 1.7805 A Sasa tafuta i_3: i_3 = (-25 + 9i_2)/(13) = (-25 + 9(410/483))/(13) = (-25 + 3690/483)/(13) = ((-12075 + 3690)/483)/(13) = (-8385)/(6279) ≈ -1.3354 A Step 3: Bainisha mikondo I_1 na I_2 kama ilivyoonyeshwa kwenye mchoro. Mkondo I_2 umeonyeshwa ukielekea kushoto kupitia kipingamizi cha 5. Kwa kutumia mikondo ya kitanzi, mkondo unaoelekea kushoto ni i_2 - i_1. I_2 = i_2 - i_1 = (410)/(483) - (6880)/(3864) = (3280)/(3864) - (6880)/(3864) = (-3600)/(3864) I_2 ≈ -0.9317 A Ishara hasi inaonyesha kuwa mkondo halisi unapita kinyume na mwelekeo wa mshale wa I_2 kwenye mchoro (yaani, kuelekea kulia). Mkondo I_1 umeonyeshwa ukielekea chini kupitia kipingamizi cha 9. Kwa kutumia mikondo ya kitanzi, mkondo unaoelekea chini ni i_2 - i_3. I_1 = i_2 - i_3 = (410)/(483) - ((-8385)/(6279)) = (410)/(483) + (8385)/(6279) I_1 = (5330)/(6279) + (8385)/(6279) = (13715)/(6279) I_1 ≈ 2.1842 A Ishara chanya inaonyesha kuwa mkondo halisi unapita katika mwelekeo sawa na mshale wa I_1 kwenye mchoro. Mikondo katika kila kipingamizi ni: I_1 = 2.18 A I_2 = -0.93 A 3 done, 2 left today. You're making progress.