The sides of a square shaped frame, Shown in the diagram are made of wires of resistances `r_(1)` and `r_(2)` The frame is places in a uniform magnetic field, which is perpendicular to the plane of figure and is increasing at a constant rate. The resistance that should be connected across the diagonal of frame, so that this wire is warmed up at greatest rate is given by `R=(n r_(1)r_(2))/(r_(1)+r_(2))`. Find n.

Answer

2

Explanation

`RI+2r_(1)I_(1)=V`
`RI-2r_(2)(I-I_(1))=-V`
`implies I=(V(r_(1)-r_(2)))/(R(r_(1)+r_(2))+2r_(1)r_(2))`
`P=I^(2)R=(V^(2)(r_(1)-r)^(2))/(R(r_(1)+r_(2))^(2)+(2r_(1)r_(2))^(2)/(R )+"constant term")`
The fastest warming occurs when
`R(r+r_(1))^(2)+((2r_(1)r_(2))^(2))/(R )` is minimum
`implies R =(2r_(1)r_(2))/(r_(1)+r_(2))`