Combination of Resistors and Galvanometer

Series Combination of Resistors

Resistances are said to be connected in series when they are joined end-to-end, so that same current flows through each of them.

Suppose three resistors R₁, R₂ and R₃ connected in series between two points A and B as shown in the figure. When the combination is connected to two terminals of a battery, the same current flows through each resister. But the potential difference across the individual resisters is different across R₁, R₂ and R₃ respectively. From Ohm’s law

\begin{align*} V_1 &= IR_1, \\ V_2 &= IR_2 \\ V_3 &= IR_3 \\ \text {If V is the total potential difference across the combination, then} \\ V &= V_1 + V_2 + V_3 \\ &= IR_1 +IR_2+ IR_3 \\ &= I(R_1+ R_2 +R_3) \\ \text {or,} \: \frac {V}{I} &= (R_1+ R_2 +R_3) \dots (i) \\ \text {Let } R_s \text {be the equivalent resistance of the series combination. Then,} \\ V &= IR_s \\ \text {or,}\: \frac {V}{I} &= (R_1+ R_2 +R_3) \\ \therefore \: R_s &= R_1+ R_2 +R_3 \dots (ii) \\ \end{align*}

Parallel Combination of Resistors

Resistances are said to be connected in parallel if one end of each resistor is connected to a common point and the other ends to another common point so that the potential difference across each resistor is the same.

Suppose three resistors R₁, R₂ and R₃ connected in parallel across two points A and B as shown in the figure. When a battery is connected across this combination, all resistors have the same potential difference. Since the resistors have different resistance, the current through each resistor is different. Let V be the potential different difference across A and B, and I₁, I₂ and I₃ be the current passing through R₁, R₂ and R₃ respectively.

Total current, I flowing in the circuit is the sum of the current in different resistors in the combination.

\begin{align*} I &= I_1 + I_2 + I_3 \dots (i) \\ \text {But from Ohm’s law,} \\ V &= I_1R_1 \\ \text {or,}: I_1 &= \frac {V}{R_1} \\ I_2 &= \frac {V}{R_2} \\ \text {and} \: I_3 &= \frac {V}{R_3} \\ \text {Substituting these values of values of currents equation} \: (i), \text {we get} \\ I &= I_1 + I_2 + I_3 \\ &= \frac {V}{R_1} + \frac {V}{R_2} + \frac {V}{R_3} \\ &= V \left ( \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3} \right ) \\ \frac {1}{V} &= \left ( \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3} \right ) \\ \text {If } \: R_e \: \text {is the equivalent resistance in parallel combination, then} \\ \frac {1}{R_e} &= \frac {I}{V} = \left ( \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3} \right ) \\ \therefore \frac {1}{R_e} &= \frac {1}{R_1} + \frac {1}{R_2} + \frac {1}{R_3} \dots (i) \\ \text {The equation } \: (ii) \: \text {can be written as} \\ \frac {1}{R_e} &= \frac {1}{R_1} + \frac {1}{R_2} \\ &= \frac {R_1 + R_2}{R_1R_2} \\ \therefore R_e &= \frac {R_1R_2}{R_1 + R_2} \dots (iii) \\ \end{align*}

Galvanometer

A galvanometer is an instrument used to detect currents in electric circuits and its direction. A moving coil placed between two pole pieces of a magnet. When a current is passed through the coil, it experiences a torque and is deflected through an angle from its mean position.

Shunt

A small resistance connected in parallel to a galvanometer is called shunt. The value of shunt is also chosen that the desired current passes through the galvanometer and the rest it through it.

In a shunt S is connected in parallel to the galvanometer. Let G be the galvanometer resistance. The equivalent resistance, R of the circuit between A and B is given by

\begin{align*} \frac {1}{R} &= \frac {1}{G} + \frac {1}{S} \\ \text {or} \: R &= \frac {GS}{G + S} \\ \text {If I is the total current passing through the circuit, then p.d. across A and B} \\ V_{AB} = IR = \frac {IGS}{G + S} \\ \text {Current passing through galvanometer,} \: I_g &= \frac {V_{AB}}{G} \\ &= \frac {IS}{G + S} \\ \text {Current passing through shunt,}\: I_s &= \frac {V_{AB}}{S} \\ &= \frac {IG}{G + S} \\ \end{align*}

Uses of Shunt

A shunt in a circuit is used to

1. Convert a galvanometer into an ammeter.
2. Increase the current in a circuit.
3. Increase the range of galvanometer.

Reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.