December 20, 2022 – Physicspupil

Three identical resistors, a battery of negligible internal resistance, and an ideal voltmeter are connected to form Circuits (a) and (b) respectively.

Given that the voltmeter reading is $8\,\mathrm{V}$ in Circuit (a), what is the voltmeter reading in Circuit (b)?

Roughwork.

Redrawing a labelled diagram for (a),

and noting that

$\begin{aligned} I_1=I_2 & = I/2 \\ I_3 & = 0 \\ I_4 & = I \\ R_\textrm{V}\parallel R & = R \\ R\parallel R & = R/2 \\ R_{\textrm{eq}} & = 3R/2 \\ \end{aligned}$

we are about to write

$\begin{aligned} I_4R & =8\,\mathrm{V}\textrm{ is given} \\ \mathcal{E} & =IR_{\textrm{eq}}\\ & = I\bigg(\frac{3R}{2}\bigg) \\ & = \frac{3}{2}(8) \\ & = 12\,\mathrm{V} \\ \end{aligned}$

and obtain that the battery has an emf $\mathcal{E}$ of $12\,\mathrm{V}$ . Likewise for (b),

where

$\begin{aligned} I_2 & = I_4 \\ I_3 & = 0 \\ R_\textrm{V}\parallel R & = R \\ R+R_\textrm{V}\parallel R & = 2R \\ R\parallel (R+R_\textrm{V}\parallel R) & = 2R/3 \\ 12= \mathcal{E} & = I\bigg(\frac{2R}{3}\bigg) \\ \Longrightarrow\enspace IR & = 18 \\ \end{aligned}$

$\begin{aligned} & \quad\enspace \begin{cases} I_1R =I_2(2R) \\ I_1+I_2 = 18/R \\ \end{cases} \\ & \Longrightarrow \begin{cases} I_1 = 12/R\\ I_2 = 6/R\\ \end{cases} \end{aligned}$

being now asked for $V=I_4R$ , the voltmeter reading, it is left to the reader.

Two lines $L_1:x+y-5=0$ and $L_2:2x-3y=0$ intersect at a point $A$ . Find the equations of the two lines passing through $A$ whose distances from the origin are equal to $2$ .