 ## Combination of Resistors and Galvanometer

Subject: Physics

#### Overview

Resistances are said to be connected in series when they are joined end-to-end, so that same current flows through each of them. This note provides us an information on a combination of resistance 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.

##### Things to remember

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

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.

A galvanometer is an instrument used to detect currents in electric circuits and its direction.

A small resistance connected in parallel to a galvanometer is called shunt.

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