Question

612
B
35. Calculate the effective
resistance between the points A
A and B in the circuit shown
in figure.
​

Answer 1

Answer:

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Answer 2

If two resistors $\sf{R_1}$ and $\sf{R_2}$ are in series connection, then their effective resistance is,

$\longrightarrow\sf{R_s=R_1+R_2}$

If two resistors $\sf{R_1}$ and $\sf{R_2}$ are in parallel connection, then their effective resistance is,

$\longrightarrow\sf{R_p=\dfrac{R_1R_2}{R_1+R_2}}$

In the fig., two groups of resistors, having resistance $\sf{6\ \Omega}$ each connected in parallel, are connected in series.

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\multiput(0,0)(35,0){2}{\put(0,0){\line(1,0){5}}\put(5,0){\line(1,1){5}}\multiput(0,0)(0,-10){2}{\put(10,5){\line(1,0){3}}\qbezier(13,5)(13.5,5.5)(14,6)\multiput(14,6)(4,0){3}{\qbezier(0,0)(1,-1)(2,-2)\qbezier(2,-2)(3,-1)(4,0)}\qbezier(26,6)(26.5,5.5)(27,5)\put(27,5){\line(1,0){3}}}\put(30,5){\line(1,-1){5}}\put(5,0){\line(1,-1){5}}\put(30,-5){\line(1,1){5}}}\put(70,0){\line(1,0){5}}\put(-5,-2){\sf{A}}\put(77,-2){\sf{B}}\put(18,10){\textcircled{\footnotesize\textsf{1}}}\put(18,-12){\textcircled{\footnotesize\textsf{2}}}\put(53,10){\textcircled{\footnotesize\textsf{3}}}\put(53,-12){\textcircled{\footnotesize\textsf{4}}}\multiput(10,8)(50,0){2}{\multiput(0,0)(0,-18){2}{ \sf{6\ \Omega} }}\end{picture}$

Here the group of resistors $\textcircled{\footnotesize\sf{1}}$ and $\textcircled{\footnotesize\sf{2}}$ are connected in parallel each with a resistance of $\sf{6\ \Omega.}$. Hence their equivalent resistance is,

$\longrightarrow\sf{R_{12}=\dfrac{6\ \Omega\cdot6\ \Omega}{6\Omega+6\Omega}}$

$\longrightarrow\sf{R_{12}=\dfrac{36\ \Omega^2}{12\ \Omega}}$

$\longrightarrow\sf{R_{12}=3\ \Omega}$

Similarly, resistors $\textcircled{\footnotesize\sf{3}}$ and $\textcircled{\footnotesize\sf{4}}$ are also connected in parallel. Hence their equivalent resistance is also,

$\longrightarrow\sf{R_{34}=3\ \Omega}$

Now the circuit can be shortened to as the following.

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\multiput(0,0)(35,0){2}{\put(0,0){\line(1,0){5}}\put(5,0){\line(1,0){5}}\multiput(0,-5)(0,-10){1}{\put(10,5){\line(1,0){3}}\qbezier(13,5)(13.5,5.5)(14,6)\multiput(14,6)(4,0){3}{\qbezier(0,0)(1,-1)(2,-2)\qbezier(2,-2)(3,-1)(4,0)}\qbezier(26,6)(26.5,5.5)(27,5)\put(27,5){\line(1,0){3}}}\put(30,0){\line(1,0){5}}}\put(70,0){\line(1,0){5}}\put(-5,-2){\sf{A}}\put(77,-2){\sf{B}}\put(18,5){\textcircled{\footnotesize\textsf{1}}}\put(18,-8){\textcircled{\footnotesize\textsf{2}}}\put(53,5){\textcircled{\footnotesize\textsf{3}}}\put(53,-8){\textcircled{\footnotesize\textsf{4}}}\multiput(8,-5)(53,0){2}{ \sf{3\ \Omega} }\end{picture}$

Hence the effective resistance between points A and B is,

$\longrightarrow\sf{R=R_{12}+R_{34}}$

$\longrightarrow\sf{R=3\ \Omega+3\ \Omega}$

$\longrightarrow\sf{\underline{\underline{R=6\ \Omega}}}$

Thus equivalent resistance is $\bf{6\ \Omega.}$