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प्रश्न
In the figure, calculate equivalent resistance between points (a) A and B (b) B and C (c) A and C.
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उत्तर
(a) Resistances of 4 Ω and 12 Ω are connected in parallel between points A and B.
Let \(\mathrm{R}_{p_1}\) be their equivalent resistance.
\[ \frac{1}{\mathrm{R}_{p_1}} = \frac{1}{4} + \frac{1}{12} \]
\[ = \frac{3 + 1}{12} \]
\[ = \frac{4}{12} \]
\[ = \frac{1}{3} \]
\[ \mathrm{R}_{p_1} = 3 \ \Omega\]
(b) Resistances of 12 Ω, 24 Ω and 8 Ω are connected in parallel between points B and C.
Let \(\mathrm{R}_{p_2}\) be their equivalent resistance.
\[ \frac{1}{\mathrm{R}_{p_2}} = \frac{1}{12} + \frac{1}{24} + \frac{1}{8} \]
\[ = \frac{2 + 1 + 3}{24} \]
\[ = \frac{6}{24} \]
\[ = \frac{1}{4} \]
\[ \mathrm{R}_{p_2} = 4 \ \Omega\]
(c) Now, equivalent resistance \(\mathrm{R}_{p_1}\) is in series with equivalent resistance \(\mathrm{R}_{p_2}\). Therefore, equivalent resistance between A and C \(\left(\mathrm{R}_{s}\right)\) is :
\[ \mathrm{R}_{s} = \mathrm{R}_{p_1} + \mathrm{R}_{p_2} \]
\[ = 3 + 4 \]
\[ = 7 \ \Omega\]
