Question

Two resistors of resistances R_1 and R_2, with R_2 > R_1, are connected to a voltage source with voltage V_0. When the resistors are connected in series, the current is I_s. When the resistors are connected in parallel, the current I_p from the source is equal to 10I_s.Let r be the ratio R_1/R_2. Find r.

Answer 1

Given two resistors :
`$R_1$` and
`$R_2$` where
`$R_2$` >
`$R_1$`

When the resistors are connected in series :

Current,
`$I_s=(\epsilon)/(R_1 + R_2)$` .............(1)

When the resistors are connected in series :

`$10I_s=(\epsilon)/(\left((R_1R_2)/(R_1+R_2)\right))$`

`$10I_s=(\epsilon(R_1+R_2))/(R_1R_2)$` ..................(2)

Therefore, dividing equation (2) by (1), we get

`$10=((R_1+R_2)^2)/(R_1R_2)$`

Now, since
`$r=(R_1)/(R_2)$` , we have
`$R_1=rR_2$`

∴
`$10=((1+r)^2)/(r)$`

`$\Rightarrow 1+r^2+2r=10r$`

`$\Rightarrow r^2-8r+1=0$`

Solving, we get, r = 0.127