Experiment: resistance and figure of merit of a galvanometer

What is this experiment?

The aim is to find the resistance G of a galvanometer by the half-deflection method, and its figure of merit k, the current per scale division.

Both the JEE Main 2027 and the NEET (UG) 2026 syllabuses list this under Unit 20, Experimental Skills: “Resistance and figure of merit of a galvanometer by half deflection method.” The NCERT textbook gives the galvanometer’s theory (Section 4.10, pages 129–132) but not the laboratory method; this page follows the NCERT Laboratory Manual (Physics, Class 12), Experiment 6, pages 53–58.

Half-deflection method

Use a zero-corrected, linear deflection scale and a stable low-voltage laboratory source. The half-deflection result requires R > S > 0 and negligible source internal resistance; if that resistance is r, replace R by R + r. Here k denotes current per division, not the torsional spring constant k used in the galvanometer theory lesson.

Circuit. A cell of emf ε, a key K₁, a high resistance box R and the galvanometer G are joined in series. A low resistance box S, the shunt, is joined across G through a second key K₂ (NCERT Laboratory Manual, Experiment 6, page 53, Fig. E 6.1).

Step 1. With K₂ open, take out a high resistance from R (a few kilohms) and close K₁. Adjust R until the deflection θ is large but on the scale, and an even number of divisions. A large θ keeps the reading error, about half a division, a small fraction of it. The current is I₁ = ε/(R + G), taking the cell’s internal resistance as negligible (Ohm’s law).

Step 2. Close K₂ and, keeping R fixed, adjust S until the deflection falls to θ/2. Now G and S are in parallel, and the current through G is half its earlier value.

Result. Without the shunt, I₁ = ε/(R + G). With the shunt, G and S in parallel have a combined resistance GS/(G + S), so the cell drives I = ε/[R + GS/(G + S)], and the share through G is IS/(G + S) = εS/[R(G + S) + GS]. Setting this equal to I₁/2 gives 2S(R + G) = R(G + S) + GS, that is RG = RS + GS, or G(R − S) = RS (NCERT Laboratory Manual, Experiment 6, Eqs. (E 6.2)–(E 6.9), pages 54–55):

• When R is much larger than S, G is close to S. That is why R is kept large: then adding the shunt hardly changes the total current, and nearly half of it goes through each branch.
• The error in taking G = S follows from the formula: G − S = S²/(R − S), so (G − S)/G = S/R. With R = 100S, the shortcut is 1 % low (reasoned from G = RS/(R − S)).
• Repeat for several values of R and take the mean of G.

Figure of merit: current per division

• The figure of merit k is the current that gives a deflection of one division. From Step 1, a current ε/(R + G) gives θ divisions, so:

• Its unit is ampere per division. It is the reciprocal of the current sensitivity in divisions per ampere: a smaller k means a more sensitive galvanometer (reasoned from NCERT Eq. (4.27), page 130).
• The emf ε of the cell must be known, for example estimated with a high-resistance voltmeter across the otherwise open cell, allowing for any residual loading, and the cell should keep a steady emf while the readings are taken (NCERT Laboratory Manual, Experiment 6, pages 56–57).
• Repeat for several values of R; k should come out the same each time, since θ is proportional to the current (NCERT Eq. (4.26)).
• Precautions: keep all connections tight; start with the largest resistance in R, so that the galvanometer is not overloaded; take deflections of even numbers of divisions, so that θ/2 is a whole number; do not change R between Step 1 and Step 2 (NCERT Laboratory Manual, Experiment 6, pages 56–57).

Solved examples

Common mistakes to avoid

Quick revision summary

• Circuit: cell, key, high resistance R and galvanometer G in series; shunt S across G through a second key.
• Without the shunt: deflection θ for I₁ = ε/(R + G). With the shunt adjusted for θ/2: G = RS/(R − S), close to S when R ≫ S.
• Figure of merit k = ε/[(R + G)θ], the current per division; the current sensitivity is 1/k.
• Keep R large and fixed, use even deflections, and repeat for several R.
• Full-scale current = k × number of divisions; it sets the shunt or series resistance for conversion.

Questions students ask

Is the half-deflection method in NCERT, and is it in the JEE Main and NEET syllabus?

The half-deflection method is not in the NCERT Class 12 Physics textbook, which gives only the galvanometer’s theory (Section 4.10, pages 129–132), but it is Experiment 6 of the NCERT Laboratory Manual Physics Class XII (pages 53–58), and both syllabuses list it: item 12 of Unit 20, Experimental Skills, of JEE Main and NEET (UG) 2026 reads “Resistance and figure of merit of a galvanometer by half deflection method.” Converting the galvanometer into an ammeter or a voltmeter, the Laboratory Manual’s next experiment, is not in that list, though the conversion itself is in Unit 13 (see conversion to an ammeter).

Can a galvanometer’s resistance be found with a one-third deflection instead of a half?

Yes, a galvanometer’s resistance can be found from any fraction of the first deflection: if the shunt S brings the deflection down to a third, with R unchanged, the same working as for a half gives G = 2RS/(R − 2S), close to 2S when R is large (beyond the NCERT text, reasoned from Ohm’s law; the NCERT Laboratory Manual Physics Class XII sets this question, Experiment 6, page 58). For n > 1 and R > (n − 1)S, a fall to 1/n of the deflection gives G = (n − 1)RS/[R − (n − 1)S]. For example, with R = 8000 Ω, a fall to one third at S = 40 Ω gives G = 2 × 8000 × 40/7920 ≈ 80.8 Ω.

How are G and the figure of merit found from a graph of 1/θ against R?

A graph of 1/θ against the series resistance R, from the readings without the shunt, is a straight line of slope k/ε that cuts the R-axis at R = −G, because ε/(R + G) = kθ gives 1/θ = (k/ε)(R + G) (beyond the NCERT text, reasoned from the figure-of-merit relation; the NCERT Laboratory Manual Physics Class XII suggests this graph, with R along the x-axis, Experiment 6, page 58). So k is the slope times ε, and extending the line back to where 1/θ would be zero gives G without any shunt, a check on the half-deflection value. A fit can reduce random scatter but does not remove systematic errors; the distant intercept may have a large uncertainty, and the reciprocal transformation gives unequal uncertainties. If source resistance r is significant, the R-axis intercept is −(G + r).

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. In a half-deflection experiment R = 6000 Ω and the half deflection comes at S = 60 Ω. Find G.

   Show answerHide answer

   G = RS/(R − S) = 6000 × 60/5940 ≈ 60.6 Ω, close to S = 60 Ω since R ≫ S.

2. Q2. Why should the deflection taken in Step 1 be an even number of divisions?

   Show answerHide answer

   So that half of it is a whole number of divisions, which can be read without estimating between marks.

3. Q3. A galvanometer has a figure of merit of 10 µA per division. What current gives a deflection of 25 divisions?

   Show answerHide answer

   I = 25 × 10 µA = 250 µA = 2.5 × 10⁻⁴ A.