Current Electricity

9.1 Introduction

• Std XI dealt with the origin of electrical conductivity and with combining resistances in series and in parallel.
• A circuit with several complex connections cannot always be reduced to one loop by series and parallel rules alone — for instance a Wheatstone network, where no two resistors are purely in series or purely in parallel.
• Gustav Robert Kirchhoff (1824-1887) framed two rules that analyse any such network. This chapter is those two laws and three instruments built on them: the Wheatstone bridge, the potentiometer and the galvanometer.

9.2 Kirchhoff's laws — the vocabulary first

• Junction: any point in an electric circuit where two or more conductors are joined together.
• Loop (mesh): any closed conducting path in an electric network.
• Branch: any part of the network that lies between two junctions.
• The laws hold for a steady current in a network of resistors, and apply to both DC and AC circuits (for AC one uses instantaneous values).

First law (current law, junction law)

• Statement: the algebraic sum of the currents at a junction in an electrical network is zero, $\sum_{i=1}^{n}I_i = 0$, where $I_i$ is the current in the $i^{th}$ conductor at a junction having $n$ conductors.
• Sign convention: arriving currents positive, leaving currents negative. For the six-conductor junction P of Fig. 9.2, $I_1 - I_2 + I_3 + I_4 - I_5 - I_6 = 0$, i.e. $I_1+I_3+I_4 = I_2+I_5+I_6$.
• It is consistent with conservation of electric charge: charge arriving per unit time must leave per unit time, since charge cannot pile up at a junction.

Second law (voltage law, loop theorem)

• Statement: the algebraic sum of the potential differences (products of current and resistance) and the electromotive forces in a closed loop is zero, $\sum IR + \sum E = 0$.
• Sign convention for a resistor: travelling along the conventional current, the p.d. is negative; travelling against it, positive.
• Sign convention for a source: tracing within the source from the negative to the positive terminal, the emf is positive; from positive to negative, negative.
• It is consistent with conservation of energy: the energy per unit charge supplied by the sources reappears as potential differences across the elements of the loop.

Steps the board expects in a Kirchhoff numerical

• Choose some direction for each current.
• Reduce the number of variables using the first law (write the third current as the sum of two).
• Determine the number of independent loops.
• Apply the voltage law to all the independent loops.
• Solve the resulting equations simultaneously.
• If a current comes out negative, the actual conventional current flows opposite to the direction chosen — say so, do not simply drop the sign.

9.3 Wheatstone bridge

• Resistance changes with temperature, strain, humidity, displacement, liquid level and so on, so measuring resistance is a way of measuring those quantities. Measurable resistances run from a few milliohm to hundreds of megohm; a Wheatstone bridge is generally used for the range tens of ohm to hundreds of ohm.
• Developed by Charles Wheatstone (1802-1875) to measure unknown resistances; also used to calibrate voltmeters and ammeters.
• Four resistances $P$, $Q$, $R$, $S$ form the quadrilateral ABCD; a cell of emf $E$ with a key sits between A and C (A at higher potential), and a galvanometer of resistance $G$ between B and D.
• Balance means $I_g = 0$, obtained by adjusting $P$, $Q$, $R$, $S$. Balancing condition: $\dfrac{P}{Q} = \dfrac{S}{R}$, i.e. the products of opposite arms are equal, $PR = QS$.
• Applications: measuring very low resistances precisely; measuring galvanometer resistance, capacitance, inductance and impedance; with an operational amplifier, measuring temperature, strain and similar physical parameters.

9.3.1 Metre bridge

• A one-metre wire of uniform cross section stretched along a metre scale on a wooden board. Its ends are fixed under two L-shaped metallic strips, and a single central strip leaves two gaps — the left gap (unknown $X$) and the right gap (resistance box $R$).
• One galvanometer terminal goes to the central strip C, the other carries the jockey J, which makes temporary contact with the wire. A cell, key and rheostat are connected between A and B.
• The wire itself supplies the other two arms of the Wheatstone bridge, so $\dfrac{X}{R} = \dfrac{l_x}{l_R}$ and $X = \dfrac{l_x}{l_R}R$, with $l_x + l_R = 100$ cm.

Kelvin's method for the resistance of a galvanometer

• The galvanometer whose resistance $G$ is wanted goes in one gap, a known resistance box $R$ in the other.
• A suitable $R$ is chosen, the key is closed and the deflection is noted without touching the jockey; the rheostat is adjusted to give a deflection of about two thirds of the range.
• The jockey is tapped at different points until a point D is found at which the deflection does not change on contact. Since the deflection is the same with and without contact between B and D, those two points are equipotential — the bridge is balanced.
• With $l_g$ the length of wire opposite the galvanometer and $l_r = 100 - l_g$ the length opposite the box, $\dfrac{G}{R} = \dfrac{l_g}{l_r}$, so $G = \left(\dfrac{l_g}{100-l_g}\right)R$.
• Note the difference from an ordinary metre bridge: here the galvanometer is in a gap, not in the central branch, and balance is a no-change in deflection rather than a zero deflection.

Post office box and strain gauge

• A post office box is a practical Wheatstone bridge with three adjustable arms: ratio arms $P$ and $Q$ (each 10, 100 or 1000 ohm) and $R$ (1 ohm to 5000 ohm), with the unknown $X$ — usually a wire — as the fourth arm, and two tap keys $K_1$ and $K_2$. At balance $X = \dfrac{RQ}{P}$, and $\rho = \dfrac{X\pi r^2}{L}$ for a wire of length $L$ and radius $r$.
• A strain gauge is a resistor whose resistance (typically 30 to 3000 ohm) is proportional to the strain in it. It replaces the unknown arm; $R_1 = R_2$ and $R_3$ is a variable resistor. With no force applied, the rheostat is adjusted until the voltmeter reads zero — this balance is the zero of the gauge. Stretching or compressing the gauge changes its resistance, unbalances the bridge, and the voltmeter reading measures the strain; more strain gives a larger reading.

9.4 Potentiometer

• An ideal voltmeter would have infinite resistance and draw no current, so that it does not disturb the p.d. it is measuring. No real voltmeter can. A potentiometer draws no current from the source at balance and therefore acts as an ideal voltmeter; it is used for accurate measurement of potential difference and of emf.

9.4.1 Principle

• Principle: the potential difference between two points on the wire is directly proportional to the length of wire between them, provided (i) the wire is of uniform cross section, (ii) the current through the wire is the same, and (iii) the temperature of the wire remains constant.
• $I = \dfrac{E}{R+r}$, $V_{AB} = \dfrac{ER}{R+r}$, and the potential gradient is $K = \dfrac{V_{AB}}{L} = \dfrac{ER}{L(R+r)}$.
• Potential gradient: potential difference per unit length of wire. For a point C at distance $l$ from A, $V_{AC} = Kl$.
• $K$ depends on the emf of the auxiliary (driving) cell, the resistance and length of the potentiometer wire, its internal and series resistances — that is, on the current through the wire and on the resistance per unit length. It does not depend on the cell being tested.
• Measuring $K$: balance a cell of known emf $E$ at length $l$; then $K = \dfrac{E}{l}$. Equivalently, $K$ is the p.d. across the whole wire divided by its length.

9.4.2 Uses

• Comparing emfs, cells connected individually: $\dfrac{E_1}{E_2} = \dfrac{l_1}{l_2}$.
• Comparing emfs, sum and difference method: $E_1+E_2 = Kl_1$, $E_1-E_2 = Kl_2$, hence $\dfrac{E_1}{E_2} = \dfrac{l_1+l_2}{l_1-l_2}$.
• Sum method: the negative terminal of the first cell is joined to the positive terminal of the second, so the emfs add to $E_1+E_2$. Difference method: like terminals are joined together, so the emfs oppose and the effective emf is $E_1-E_2$ (with $E_1>E_2$). The difference connection is not a parallel combination of cells.
• Internal resistance of a cell: $r = R\left(\dfrac{l_1-l_2}{l_2}\right)$, where $l_1$ balances the emf (box open) and $l_2$ the terminal p.d. (box $R$ closed).
• Practical applications: as a voltage divider, to change the output voltage of a supply continuously, the voltage dividing in the ratio of the lengths $l_1$ and $l_2$; as audio controls, the sliding faders and rotary knobs used for frequency attenuation and loudness control; as a motion sensor, when the slider is attached to a moving part, so that the change in potential (amplified) is calibrated in terms of displacement.
• Internal resistance of a cell itself depends on the nature and concentration of the electrolyte, the separation between the electrodes, the area of the electrodes dipped in the electrolyte, and the temperature.

9.4.3 Potentiometer versus voltmeter

• Merits, in the board's own words: more sensitive than a voltmeter; can measure both p.d. and emf, whereas a voltmeter draws current and so always reads terminal p.d.; very accurate, with a least count far better than a voltmeter's.
• Demerits: it is not portable, and direct measurement of p.d. or emf is not possible — every reading is a balancing length that must then be converted.

9.5 Galvanometer

• A galvanometer detects weak electric currents. Its coil is pivoted (or suspended) between the concave pole faces of a strong laminated horseshoe magnet; current through the coil deflects it, and the deflection is proportional to the current and is read off by a pointer on a scale.
• The coil has moderate resistance (about 100 ohm) and a small current capacity (about 1 mA), so on its own it can neither measure large currents nor withstand them.

9.5.1 As an ammeter

• Three changes needed: increase its effective current capacity; decrease its effective resistance (an ideal ammeter has zero resistance, since a finite $G$ in series would itself reduce the current being measured); protect it from damage by excessive current.
• All three are achieved by a shunt $S$, a low resistance in parallel with the galvanometer.
• Uses of a shunt: it diverts a large part of the total current through an alternative path and so protects the instrument; it increases the range of the ammeter; it decreases the resistance between the points to which it is connected.
• $S = \left(\dfrac{I_g}{I-I_g}\right)G$, and for an $n$-fold increase in range, $S = \dfrac{G}{n-1}$ with $n = \dfrac{I}{I_g}$. Fraction through the shunt: $\dfrac{I_s}{I} = \dfrac{G}{S+G}$.

9.5.2 As a voltmeter

• Three changes needed: increase its voltage measuring capacity; increase its effective resistance (an ideal voltmeter has infinite resistance, since a low-resistance meter draws current and lowers the very p.d. being measured); protect it from excess applied p.d.
• All three are achieved by a high resistance $X$ in series with the galvanometer.
• $V = I_gX + I_gG$, hence $X = \dfrac{V}{I_g} - G$, and in terms of the range multiplier, $X = G(n_v-1)$ with $n_v = \dfrac{V}{I_gG}$.

Thermoelectricity (Seebeck effect)

• A resistor converts electrical energy into thermal energy. The reverse conversion, thermal energy directly into electrical energy, was discovered by Seebeck and is called the thermoelectric effect.
• If two different metals are joined to form a closed loop and the two junctions are kept at different temperatures, a small emf (thermo emf) is produced and current flows. The pair of dissimilar metals is a thermocouple.
• Direction: in an antimony-bismuth couple the current flows from antimony to bismuth at the cold junction (the ABC rule); in a copper-iron couple it flows from copper to iron at the hot junction.
• The effect is reversible — interchanging the hot and cold junctions reverses the current.
• With the cold junction at $0\,^\circ\text{C}$ and the hot junction at $T\,^\circ\text{C}$, $E = \alpha T + \dfrac{1}{2}\beta T^2$, where $\alpha$ and $\beta$ are the thermoelectric constants; the graph of $E$ against $T$ is therefore a parabola.

Standard results worth memorising

• At the balance point of a potentiometer, current is drawn from the auxiliary (driving) battery only — none from the cell being tested.
• Kirchhoff's first law is a statement of conservation of charge; the second law, of conservation of energy.
• If the current through the potentiometer wire is increased, $K$ increases, so the balancing length $l = \dfrac{E}{K}$ decreases and the null point shifts towards A. If the current is decreased, $K$ decreases and the null point shifts away from A, towards B — and if $K$ falls below $\dfrac{E}{L}$, no null point is obtained at all.
• A resistance $s$ shunted across one arm to balance a bridge: for arms 10, 10, 10 and 15 ohm, the 15 ohm arm must read 10 ohm, so $\dfrac{15s}{15+s} = 10$ gives $s = 30\ \Omega$.
• A ring or bangle connected in a gap at two diametrically opposite points contributes two halves in parallel, i.e. $\dfrac{R}{4}$ where $R$ is the resistance of the whole ring.
• A circular loop of total resistance $R$ tapped at two points a quarter of the circumference apart offers $\dfrac{R}{4}$ and $\dfrac{3R}{4}$ in parallel, i.e. $\dfrac{3R}{16}$.
• A voltmeter of resistance $R_V$ across a cell of emf $E$ and internal resistance $r$ reads $V = \dfrac{ER_V}{R_V+r}$, always less than $E$.

Traps that cost marks

• Writing the balancing condition as $\dfrac{P}{Q} = \dfrac{R}{S}$. The textbook's labelling gives $\dfrac{P}{Q} = \dfrac{S}{R}$. Safest is to remember that the products of opposite arms are equal.
• Forgetting that in the metre bridge $l_x$ and $l_R$ are lengths of the same wire, so $\rho$ and $A$ cancel; students sometimes try to substitute a resistivity that was never given.
• Mixing up $l_g$ in Kelvin's method: it is the length of wire on the side of the galvanometer, and $100-l_g$ is on the side of the box. Interchanging them inverts the answer.
• In the internal resistance formula $r = R\left(\dfrac{l_1-l_2}{l_2}\right)$, $l_1$ is the longer, open-circuit length. Getting $l_1$ and $l_2$ the wrong way round gives a negative $r$.
• Writing $\dfrac{E_1}{E_2} = \dfrac{l_1-l_2}{l_1+l_2}$ in the sum and difference method. The sum length $l_1$ is the larger one and belongs on top: $\dfrac{E_1}{E_2} = \dfrac{l_1+l_2}{l_1-l_2}$.
• Using $X = \dfrac{V}{I_g}$ and forgetting to subtract $G$ when converting a galvanometer into a voltmeter.
• Substituting $I_g$ in mA or $\mu$A into $S = \left(\dfrac{I_g}{I-I_g}\right)G$ while $I$ is in A. Convert both to the same unit first.
• Confusing $n = \dfrac{I}{I_g}$ (ammeter) with $n_v = \dfrac{V}{I_gG}$ (voltmeter); $S = \dfrac{G}{n-1}$ is a division, $X = G(n_v-1)$ a multiplication.
• Dropping the minus sign in a Kirchhoff answer. A negative current is a correct answer with a physical meaning — the current flows opposite to the assumed direction — and the board expects that sentence.
• Saying the potentiometer "draws a very small current" at balance. At balance it draws exactly zero current from the cell being tested; that is the whole point.
• Sliding the jockey instead of tapping it, and being asked why not: the wire gets scraped, its cross section is no longer uniform, and the null point becomes unreliable.
• Claiming a voltmeter can measure emf. It measures terminal p.d. only, because it always draws some current.