☰ خطة العام · الفيزياء (بالإنجليزية)

From galvanometer to ammeter, voltmeter and ohmmeter

الأسبوع 7 · الفصل الثاني (٤): تحويل الجلفانومتر إلى أميتر (مجزئ التيار) وإلى فولتميتر (مضاعف الجهد)، والأوميتر وتدريجه غير المنتظم، والأجهزة التناظرية والرقمية · 3 دروس
2

From galvanometer to voltmeter: the multiplier

Physics Sec 3 Term 1, Chapter 2: pp. 46–51
GoalFind the multiplier resistance that turns a galvanometer into a voltmeter of a chosen range.
New words
multiplier (Rm) · a large resistor connected in series with the galvanometer so that the voltage across the pair is sharedvoltmeter · a galvanometer with a multiplier, connected in parallel with the part being measured; its own resistance is very largerange of a voltmeter · the voltage V that makes the pointer reach the end of the scale
Explanation

A galvanometer can stand only a small voltage, IgRg across its coil. To measure larger voltages, a large resistor Rm is connected in series with it, so that the full-scale current Ig flows when the voltage across the pair is V: V = Ig(Rg + Rm), which gives Rm = V ÷ Ig − Rg. The larger the range V, the larger the multiplier. The resistance of the whole voltmeter is V ÷ Ig, very large, because a voltmeter is connected across the part to be measured, with its positive terminal on the side of the higher potential, and it should take almost no current from the circuit. An ammeter goes in series; a voltmeter goes across.

AmmeterVoltmeter
extra resistorshunt Rs, very small, in parallelmultiplier Rm, large, in series
formulaRs = IgRg ÷ (I − Ig)Rm = V ÷ Ig − Rg
connectedin series with the circuitacross the part measured
resistance of the instrumentvery smallvery large
Worked examples
Rg = 60 Ω, Ig = 1.5 mA, wanted range 3 V. V ÷ Ig = 3 ÷ 0.0015 = 2000 Ω, so Rm = 2000 − 60 = 1940 Ω.
The same galvanometer for 30 V: V ÷ Ig = 30 ÷ 0.0015 = 20 000 Ω, so Rm = 20 000 − 60 = 19940 Ω. The voltmeter’s own resistance is 20 000 Ω.
Two 10 kΩ resistors are in series with a 12 V cell of negligible internal resistance, so each has 6 V. A voltmeter of resistance 10 kΩ is connected across one of them. That resistor and the voltmeter in parallel make 5 kΩ, the total is 15 kΩ, the current is 12 ÷ 15 000 = 0.8 mA, and the voltmeter reads 0.8 × 10⁻³ × 5000 = 4 V instead of 6 V. A voltmeter must have a resistance much larger than the part it measures.
Class activity

Paper practice. Draw a circuit with a cell and two resistors in series. Add an ammeter and a voltmeter to measure the current and the voltage across the second resistor. Check that the ammeter is in the main wire and the voltmeter is across the resistor, with + on the side nearer the cell’s positive pole.

Practice
1
A galvanometer has Rg = 100 Ω and Ig = 2 mA. Find the multiplier for a 5 V range.
2
The same galvanometer is to read up to 50 V. Find the multiplier and the resistance of the voltmeter.
3
A galvanometer has Rg = 50 Ω and Ig = 3 mA. Find the multiplier for a 15 V range.
4
Two 6 kΩ resistors are in series with a 9 V cell of negligible internal resistance. A voltmeter of resistance 6 kΩ is connected across one of them. What does it read?