Electricity

Blazing goIITian

Joined: 19 Mar 2012
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25 Mar 2012 10:40:38 IST
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 12 WIRES , EACH OF RESISTANCE r ARE JOINED TOGETHER TO FORM A CUBE. FIND THE EQUIVALENT RESISTANCE IN THE EDGE OF CUBE . I DID IT USING KIRCHOFF LAW  N FOUND IT TO BE VERY LONG.PLEASE GIVE ME A SHORTCUT TO SOLVE IT. I WILL BE REALLY THANKFUL!



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Hot goIITian

Joined: 10 Oct 2011
Posts: 120
25 Mar 2012 10:54:36 IST
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use symmetry to solve it thats the short cut


Blazing goIITian

Joined: 19 Mar 2012
Posts: 421
25 Mar 2012 13:44:02 IST
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I USED IT BUT DID NOT GET THE ANSWER . YOU TRY IT OUT & IF YOU GET 7r/12 , PLEASE POST THE STEPS .
edison's Avatar

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Joined: 19 Oct 2006
Posts: 7804
28 Mar 2012 11:54:40 IST
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In this circuit the vertices E and D (F and C) are equivalent, they are on the same potential. The resistance of the whole cube is not changed by merging these vertices into one. Let us merge the vertices E and D (F and C) into one junction, redraw the circuit into the plane and supplement each cube edge with a resistor. The resistance of each edge is R.

The plotting of the circuit in the plane

Let us consider each loop consisting of two resistors R in parallel connection as a single resistor whose resistance R1 is:

\frac{1}{R_1}\,=\,\frac{1}{R}+\frac{1}{R}R_1\,=\,\frac{R}{2}

Let us simplify the sketch of the circuit:

The simplyfied sketch of the circuit

The expression for the resistance R2 between the junctions (ED) and (CF) reads:

\frac{1}{R_2}\,=\,\frac{1}{R_1}+\frac{1}{\left(R_1+R+R_1\right)}\frac{1}{R_2}\,=\,\frac{1}{\frac{R}{2}}+\frac{1}{2\frac{R}{2}+R}\,=\,\frac{2}{R}+\frac{1}{2R}\,=\,\frac{5}{2R}R_2\,=\,\frac{2}{5}R

The expression for resistance of the whole cube reads:

\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{R_1+R_2+R_1}

Inserting R_1\,=\,\frac{R}{2},  R_2\,=\,\frac{2}{5}R we obtain RAB

\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{2\frac{R}{2}+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{R+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{\frac{7}{5}R}\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{5}{7R}\,=\,\frac{12}{7R}

Thus the total resistance of the cube between the vertices A and B is

R_{AB}\,=\,\frac{7}{12}R




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