Each of the resistance in the network shown in the figure is R . The resistance between A and B is
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The circuit is symmetrical about an axis perpendicular to AB.
Now Consider the "X" formed by thefour resistors in two parts i.e. upper "v"and lower "^"(inverted v). Due to circuit symmetry, the current in the resistors of upper "v" of "X" will be same. Similarly, the current in the resistors of lower "^" of "X" will besame.
So "X" will now be separated into "v"and "^". Upper v and inverted v of Xhave now been separated. Each of theses two will a resistance of (R+R)=2R.
Now you have two paths from A to B.
First path has a resistance = R + (R||2R) + R = 8R/3
Second path has a resistance = 2R||R = 2R/3
These two are in parallel, both are connected to A and B.
So equivalent resistance of network about A&B= (8R/3)||(2R/3) = 8R/15
Alternate:
Excite the network by a 1V source and use nodal analysis to find the voltages at the 3 nodes in the network(upper two and centre point will be the three nodes). Then find the total current isupplied by the source. Then find equivalent resistance of network as R_eq =1/i .
Now Consider the "X" formed by thefour resistors in two parts i.e. upper "v"and lower "^"(inverted v). Due to circuit symmetry, the current in the resistors of upper "v" of "X" will be same. Similarly, the current in the resistors of lower "^" of "X" will besame.
So "X" will now be separated into "v"and "^". Upper v and inverted v of Xhave now been separated. Each of theses two will a resistance of (R+R)=2R.
Now you have two paths from A to B.
First path has a resistance = R + (R||2R) + R = 8R/3
Second path has a resistance = 2R||R = 2R/3
These two are in parallel, both are connected to A and B.
So equivalent resistance of network about A&B= (8R/3)||(2R/3) = 8R/15
Alternate:
Excite the network by a 1V source and use nodal analysis to find the voltages at the 3 nodes in the network(upper two and centre point will be the three nodes). Then find the total current isupplied by the source. Then find equivalent resistance of network as R_eq =1/i .
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