Resistance of a Galvanometer by Half Deflection Method

Aim

To determine the resistance of a galvanometer by half-deflection method and to find its figure of merit

Apparatus and Material Required

• Moving coil galvanometer,
• Battery or a battery eliminator (0 – 6 V),
• Resistance box \(R_{B1}\) of range 0 – 10 \(k \Omega\),
• Resistance box \(R_{B2}\) of range 0 – 200 \( \Omega\),
• Two one way keys,
• Voltmeter,
• Connecting wires and a piece of sand paper

Principle

Moving Coil Galvanometer is a sensitive electrical device used to detect current. It works on the principle that “a current carrying coil placed in a uniform magnetic field experiences a torque”.

When a coil carrying current \(I\) is placed in a radial magnetic field, the coil experiences a deflection \(\theta\) which is related to \(I\) as \[I = k\theta \text{ ——————— (i)}\] where k is a constant of proportionality and is termed as figure of merit of the galvanometer.

The adjoining figure shows the circuit arrangement required for finding the resistance \(G\) of the
galvanometer by half deflection method.

Setup Explanation

When the key \(K_1\) is closed and \(K_2\) is open, the total current withdrawn from the battery (emf = \(E\)) also passes through the galvanometer. The current \(I_g\) is \[I_g = \frac{E}{R+G}\text{ ——————— (ii)}\] where \(G\) is the resistance of the galvanometer.

If \(I_g\) causes a deflection \(\theta\), then \[I_g = k\theta \text{ ——————— (iii) (from (i) )}\] \[ k\theta = \frac{E}{R+G}\text{ ——————— (from (ii) and (iii))}\]

For closed keys \(K_1\) and \(K_2\), if the deflection in the galvanometer for a some suitable shunt resistance \(S\) is half, then \[\text{deflection} = \frac{\theta}{2}\], total resistance of the circuit, \(R_{eq}\) \[R_{eq} = R + \frac{SG}{S+G}\] and total current withdrawn from the battery is \[I = \frac{E}{R_{eq}}=\frac{E}{R + \frac{SG}{S+G}}\]

If the current through galvanometer, when the circuit has the shunt resistance as shown, is \(I_{gs}\), then current through the shunt resistance is \(I -I_{gs} \) and \[I_{gs} \times G = (I -I_{gs}) \times S\]

\[I_{gs} =\frac{IS}{G+S} \]where \(I = \frac{E}{R + \frac{SG}{S+G}}\).

Solving for \(I_{gs}\), we get \[I_{gs} = \frac{ES}{R(G+S)+GS}\text{ ——————— (iv)}\]

As the deflection in the galvanometer due to the current \(I_{gs}\) in it is \(\frac{\theta}{2}\), \[I_{gs} = k \frac{\theta}{2}\text{ ——————— (v)}\]

Dividing the equation (iii) by (iv), \[\frac{I_g}{I_{gs}} = \frac{k\theta}{k\frac{\theta}{2}} =2\]

Taking the values of \(I_g\) and \(I_{gs}\) from equations (ii) and (iv), we have \[\frac{\frac{E}{R+G}}{\frac{ES}{R(G+S)+GS}} =2\]

\[\Rightarrow \frac{E(R(G+S)+GS)}{ES(R+G)} =2\]

\[\Rightarrow R(G+S)+GS =2 S(R+G)\]

\[\text{or,} RG+RS+GS =2RS+2GS)\]

\[\Rightarrow RG =RS+GS\]

\[\text{or,} RG – GS=RS\]

\[\Rightarrow G(R – S)=RS\]

\[\text{or,} G = \frac{RS}{R-S}\text{ ———— (vi)}\]

When R is very high (~ 10 \(k \Omega\)) in comparison to shunt resistance \(S\) (~100 \(\Omega\)), then \[G\simeq S\]

Finding figure of merit

Figure of merit \(k\) of a galvanometer is defined as the current required for deflecting the pointer by one division. It is \[k =\frac{I}{\theta}\]

For opend key \(K_2\), \[k = \frac{I_g}{\theta} \]

\[k = \frac {1}{\theta}\left( \frac{E}{R+G} \right) \text{ ———— (vii)}\]

By knowing the values of \(\theta\), E, R, and G, figure of merit \(k\) is calculated.

Procedure

1. Clean the connecting wires with sand paper and make neat and tight connections as per the circuit diagram.
2. From the high resistance box \(R_{B1}\), remove 5 k\(\Omega\) key and then close the key \(K_1\). Keep the key \(K_2\) open. Adjust the resistance R from this resistance box to get full scale deflection on the galvanometer dial.
3. Record the values of resistance, R and deflection \(\theta\). Take care that \(\theta\) is an even number of division.
4. Repeat the steps 2 to 4 to get five set of observations that includes \(R\), \(S\), \(\theta\), and \(\frac{\theta}{2}\). Write them in tabular form.
5. Calculate the resistance of galvanometer and figure of merit using equations (vi) and (vii).

Observations

Emf of the battery E = 4 V

Number of divisions on full scale of galvanometer = 30 divisions

Calculations

Mean value of G (resistance of galvanometer) \[G = \frac{125.4+132.8+131.9+135.2+127.1}{5}\] \[G= 130.5 \Omega\]

Mean value of k (figure of merit of galvanometer) \[k = \frac{0.0000165+0.0000170+0.0000168+0.0000171+0.0000170}{5}\] \[ k= 0.0000169 \text{ ampere/division}\] \[ k= 1.69 \times 10^{-5} \text{ ampere/division}\]

Results

Resistance of galvanometer by half-deflection method = \(\Omega\)

Figure of merit of the galvanometer, \(k\)= ampere/division

Precautions

1. All the connections and plugs in the resistance box should be tight.
2. Emf of the battery should be constant.
3. Use as high values of R as practically possible. This ensures correct value of G.
4. Insert the key \(K_1\) only after taking out high value resistance, R, from the resistance box. Otherwise, galvanometer may burn.
5. Adjust R such that deflection in galvanometer is of even division so that \(\frac{\theta\}{2}\) is more conveniently obtained.

Sources of error

1. Plugs in the resistance boxes may be loose or they may not be clean.
2. The emf of the battery may not be constant.

Conclusion

The experiment successfully determined the resistance of the galvanometer using the half- deflection method and calculated its figure of merit, which represents the current needed for unit deflection. This method proved effective and reliable, providing accurate measurements of the galvanometer’s resistance and sensitivity.

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Reference: NCERT Lab Manual