Physics · Chapter 14
Study notes aligned to the official NEB syllabus.
This chapter develops the tools needed to analyse networks that cannot be reduced by simple series and parallel combinations alone: Kirchhoff's laws, the Wheatstone bridge and its practical forms (meter bridge and potentiometer), the idea of superconductivity, and the conversion of a galvanometer into an ammeter or a voltmeter.
Recall the two combination rules used throughout:
A junction (node) is a point where three or more conductors meet; a loop (mesh) is any closed conducting path.
Statement: At any junction in an electrical network, the algebraic sum of the currents is zero; equivalently, the total current flowing into a junction equals the total current flowing out of it.
$$ \begin{aligned} \sum I &= 0 \quad\text{or}\quad \sum I_{\text{in}} \ &= \sum I_{\text{out}} \end{aligned} $$
For the currents at a junction $O$ (taking currents towards the junction as positive and away as negative):
$$ \begin{aligned} I_1 + I_2 + I_3 + I_4 - I_5 - I_6 &= 0 \quad\Longrightarrow\quad I_1 + I_2 + I_3 + I_4 \ &= I_5 + I_6 \end{aligned} $$
Basis: conservation of electric charge (charge does not accumulate at a junction in steady state).
Statement: Around any closed loop of a network, the algebraic sum of the changes in potential is zero; equivalently, the algebraic sum of the emfs equals the algebraic sum of the products of current and resistance.
$$ \begin{aligned} \sum E &= \sum IR \quad\text{or}\quad \sum \Delta V \ &= 0 \ \text{(around a closed loop)} \end{aligned} $$
Basis: conservation of energy (a unit charge taken once round a loop returns to the same potential).
Sign convention (when applying KVL):
The Wheatstone bridge is a network of four resistances used to measure an unknown resistance accurately by a null (balance) method.
Arrangement: four resistors $P$, $Q$, $R$ and $S$ form the four arms of a quadrilateral $ABCD$. A cell is connected across one diagonal ($A$ to $C$) and a sensitive galvanometer across the other diagonal ($B$ to $D$). Here $R$ is a known variable resistance and $S$ (often written $X$) is the unknown resistance.
Balance condition. The bridge is balanced when the galvanometer shows no deflection, i.e. no current flows through it ($I_g = 0$). Then $B$ and $D$ are at the same potential. Applying Kirchhoff's laws with $I_g = 0$:
Dividing the two equations:
$$\frac{P}{Q} = \frac{R}{S} \qquad\Longrightarrow\qquad \boxed{\dfrac{P}{Q} = \dfrac{R}{S}}$$
Importance: the balance condition is independent of the cell's emf and of the galvanometer resistance, so the measurement is highly accurate. It is the working principle of the meter bridge and of many resistance-based sensors (strain gauges, thermistors).
This chapter develops the tools needed to analyse networks that cannot be reduced by simple series and parallel combinations alone: Kirchhoff's laws, the Wheatstone bridge and its practical forms (meter bridge and potentiometer), the idea of superconductivity, and the conversion of a galvanometer into an ammeter or a voltmeter.
Recall the two combination rules used throughout:
A junction (node) is a point where three or more conductors meet; a loop (mesh) is any closed conducting path.
Statement: At any junction in an electrical network, the algebraic sum of the currents is zero; equivalently, the total current flowing into a junction equals the total current flowing out of it.
For the currents at a junction (taking currents towards the junction as positive and away as negative):
Basis: conservation of electric charge (charge does not accumulate at a junction in steady state).
Statement: Around any closed loop of a network, the algebraic sum of the changes in potential is zero; equivalently, the algebraic sum of the emfs equals the algebraic sum of the products of current and resistance.
Basis: conservation of energy (a unit charge taken once round a loop returns to the same potential).
Sign convention (when applying KVL):
The Wheatstone bridge is a network of four resistances used to measure an unknown resistance accurately by a null (balance) method.
Arrangement: four resistors , , and form the four arms of a quadrilateral . A cell is connected across one diagonal ( to ) and a sensitive galvanometer across the other diagonal ( to ). Here is a known variable resistance and (often written ) is the unknown resistance.
Balance condition. The bridge is balanced when the galvanometer shows no deflection, i.e. no current flows through it (). Then and are at the same potential. Applying Kirchhoff's laws with :
Dividing the two equations:
Importance: the balance condition is independent of the cell's emf and of the galvanometer resistance, so the measurement is highly accurate. It is the working principle of the meter bridge and of many resistance-based sensors (strain gauges, thermistors).