Series and Parallel Combination

Parallel Combination

Voltage across each member is same

$\Delta V=I_1 R_1=I_2 R_2=I_3 R_3$

Series Combination

Same current through each member

$V_{a b}=I\left(R_1+R_2+R_3\right)$

When switch is open

Ce : $R_1$ is in series with $R_2$

$R_{12}=12 \Omega$

$R_{34}=20 \Omega$

$R_{12}$ and $R_{34}$ are parallel

$\frac{1}{R_{eq}} =\frac{1}{R_{12}}+\frac{1}{R_{24}}$

$=\frac{1}{12}+\frac{1}{20}$

$R_{eq} =\frac{15}{2} \Omega$

$I=\frac{21}{15 / 2}=2.8 \mathrm{~A}$

$\Delta V_{c e}=\Delta V_{d f}=21 \mathrm{~V}$

$I_1=\frac{21}{12}=1.75 \mathrm{~A}$

$I_2=\frac{21}{20}=1.05 \mathrm{~A}$

$V_a=21-I_1 R_1=21-\frac{7}{4} \times 4=14 \mathrm{~V}$

$V_b=21-I_3 R_3=21-\frac{21}{20} \times 12=8.4 \mathrm{~V}$

$v_a=v_b$

$v_c=v_d$

$v_e=v_f$

$a \equiv a^{\prime}$

$b \equiv b^{\prime}$

$R_1 || R_3 \Rightarrow R_{13} =\frac{R_1 R_3}{R_1+R_3}$

$=\frac{48}{16}=3 \Omega$

$R_2 || R_4 \Rightarrow R_{24} =\frac{R_2 R_4}{R_2+R_4} =4 \Omega$

$I=3 \mathrm{~A}$

Drop across ca is 9V

Drop acros ae is 12V

Drop across $c a=9 v$

$I_1=\frac{9}{4}=2.25 \mathrm{~A}$

Drop across ae = 12

$I_2=\frac{12}{7}=1.5 \mathrm{~A}$

$I_{a b}=0.75 \mathrm{~A}$

$R_{\text {eq }}=\frac{2 \times 4}{2+4}=\frac{8}{6}=\frac{4}{3} \Omega$

Net Resistance

$R_{\text {final }}=4 \times \frac{4}{3}=\frac{16}{3} \Omega$

$R_{eq}$ is Parallel to R

$= \frac{R_{e q} \cdot R}{R_{e q}+R}=R_{e q}^{\prime}$

$=R_{e q}^{\prime}+2 R=\frac{R_{e q} R}{R_{e q}+R}+2 R$

$\frac{R_{e q} R}{R_{e q}+R}+2 R =R_{e q}$

$2 R^2+3 R R_{e q}=R_{e q}^2+R R_{e q}$

$R_{e q}^2-2 R R_{e q}-2 R^2=0$

$R_{e q} =\frac{2 R \pm \sqrt{4 R^2+8 R^2}}{2}$

$=\frac{2 R \pm 2 \sqrt{3} R}{2}=(1 \pm \sqrt{3}) R$

$R_{\text {eq }} =(1+\sqrt{3}) R$

$\rho$ Resistivity

r = a + $\frac {b-a}{L}$ x

$d R(x) =\rho \cdot \frac{d x}{\pi\left[a+\frac{b-a}{L} x\right]^2}$

$R =\frac{\rho}{\pi} \int_0^L \frac{d x}{\left[a+\frac{b-a}{L} x\right]^2}$

$=\frac{\rho}{\pi} \times \frac{b-a}{a b}=\frac{\rho}{\pi} \cdot \frac{L}{a b}$

$\rho_{A L}^0=2.75 \times 10^{-8} \Omega \mathrm{m}$

$\alpha_{A L}=.004 /{ }^{\circ} \mathrm{C}$

$\rho_c^0=5 \times 10^{-5} \Omega \mathrm{m}$

$\alpha_c=-0.0005 / ^0 \mathrm{C}$

$R_{A L}+R_C= R_{A L}^0\left(1+\alpha_{A L} \cdot \Delta T\right)$

$+R_c^0\left(1+\alpha_0 \cdot \Delta T\right)$

$R_{A L}+R_C \equiv R_{A L}^{\circ}+R^{0}_{C}$

$R_{A L} \alpha_{A L} \cdot \Delta T=-R_{c}^0 \alpha_c \cdot \Delta T$.

$\alpha_{AL} \rho_{A L}^0 L^{A L}=-\alpha_c \rho_c^{0} \cdot L^C$

$L^\mu / L^c \approx 227$

$R_4 || R_5 \Rightarrow R_{45}$

$R_2, R_3$ and $R_{45}$ are in series $R_{2345}$

$R_6$ and $R_8$ in series = $R_{68}$

$R_9 || R_{10}$ $\Rightarrow R_9^{\prime}$

• $R_9 \text{in series with} R_{68}$.

• $R_{689}{'}$ is parallel to $R_7^{\prime}$

$\Rightarrow R^{\prime}$

More than 1 battery in the circuit

Cells in series

$V_c-I r_2+\varepsilon_2-I r_1+\varepsilon_1=V_a$

$V_a-V_c=(\underbrace{\varepsilon_1+\varepsilon_2}_{\varepsilon})-I \underbrace{r_1+r_2})$

An equivalent emf $\varepsilon_1+\varepsilon_2$ and effective internal resistance

$r_1+r_2$