A uniform wire of resistance `4Omega` is bent into circle of radius `r`. As specimen of the same wire is connected along the dimeter of the circle. What is the equivalent resistance across the ends of this wire?

Answer

`(4)/(4 + pi) Omega`

Explanation

Circumference of the circuit `= 2 pi r`
its resistance `R = 4 Omega`
resistance per unit length of the wire `= (R )/(2 pi R)`
Resistance of the specimen connected along the diameter is `R_(1) = (R )/(2 pi r) xx 2 r = (R )/(pi)`
The resistance of each senicircular wire `R_(2) = (R )/(2)`
Equivalent resistance `R_(3)` across the specimen is
`(1)/(R_(3)) = (2)/(R ) + (2)/(R ) + (pi)/(R ) + (4 + pi)/(R )`
or `R_(3) = (R )/(4+ pi) = (4)/(4 + pi) Omega`