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Consider the circuit shown in figure . ...

Consider the circuit shown in figure . For a given resistance `R_(0)` what must be the value of `R_(1)` so that the equivalent resistance between the terminals is equal to `R_(0)` ?

Text Solution

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The equivalent resistance `R'`, due to the three resistors on the right is
`(1)/(R'= (1)/(R) + (1)/(R_(0))+R_(1)) = (R_(0)+2R_(1))/(R_(1)(R_(0)+R_(1)))`
or `R' = (R_(1)(R_(0)+R_(1)))/(R_(0)+2R_(1))`
Since R' is in series with the fourth resistor `R_(1)` , the equivalent resistance of the entire configuration becomes .
`R_(eq) = R_(1) + ( R_(1)(R_(0))+R_(1))/(R_(0)+2R_(1))= (3R_(1)^(2) + 2R _(1) R_(0))/(R_(0)+ 2R_(1))`
If `R_(eq) = R_(0)` then
`R_(0)(R_(0)+2R_(1))+ 3R_(1)^(2) + 2R_(1)R_(0)implies R_(0)^(2) = 3R_(1)^(2)`
or `R_(1) = (R_(0))/(sqrt(3))`
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