Class 12 Physics Notes Chapter 6 (Chapter 6) – Lab Manual (English) Book

Alright students, let's focus on Experiment 6 from your Physics Lab Manual. This experiment is quite important, not just for your practical exams, but also because the concepts involved frequently appear in competitive government exams. We'll break down the determination of a galvanometer's resistance using the half-deflection method and finding its figure of merit.

Experiment 6: To determine the resistance of a galvanometer by the half-deflection method and to find its figure of merit.

1. Aim:

• To find the resistance (G) of a given galvanometer using the half-deflection method.
• To determine the figure of merit (k) of the same galvanometer.

2. Apparatus Required:

• A Weston type galvanometer
• A battery or accumulator (cell) with a stable EMF (E)
• Two resistance boxes (one high resistance, R, typically 0-10,000 Ω; one low resistance, S, typically 0-100 Ω, used as shunt)
• Two one-way keys (K₁, K₂)
• Connecting wires (copper wires)
• Sandpaper (for cleaning wire ends)
• A voltmeter (to measure the EMF of the battery, optional if EMF is known)

3. Theory:

(a) Resistance of Galvanometer (G) by Half-Deflection Method:

• Galvanometer: A device used to detect small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque, causing deflection.

• Principle of Half-Deflection:

  1. A high resistance 'R' is connected in series with the galvanometer (G) and a cell (E). The key K₁ is closed. The resistance R is adjusted so that the galvanometer shows a large deflection, say θ (preferably an even number of divisions).
     The current flowing through the galvanometer is:
     I<0xE1><0xB5><0x8D> = E / (R + G) (Assuming the cell has negligible internal resistance and R >> G)
     Since deflection θ is proportional to the current, θ ∝ I<0xE1><0xB5><0x8D>
     So, θ ∝ E / (R + G) ---(1)

  2. Now, a low resistance 'S' (shunt resistance) is connected in parallel across the galvanometer, and key K₂ is also closed. The value of S is adjusted such that the deflection in the galvanometer reduces to exactly half of the previous value, i.e., θ/2.
     When the shunt S is connected, the combination of G and S in parallel has an equivalent resistance G' = (G * S) / (G + S).
     The total resistance in the circuit is now R + G' = R + [(G * S) / (G + S)].
     The total current from the battery is I' = E / [R + (GS / (G+S))].
     This current I' divides between G and S. The current through the galvanometer (I<0xE1><0xB5><0x8D>') is given by the current divider rule:
     I<0xE1><0xB5><0x8D>' = I' * [S / (G + S)]
     I<0xE1><0xB5><0x8D>' = { E / [R + (GS / (G+S))] } * [S / (G + S)]
     I<0xE1><0xB5><0x8D>' = E * S / [R(G+S) + GS]

     The new deflection is θ/2. So, θ/2 ∝ I<0xE1><0xB5><0x8D>'
     θ/2 ∝ E * S / [R(G+S) + GS] ---(2)

  3. Approximation: If R is very large compared to G (and therefore also large compared to S and G'), the introduction of the shunt S changes the total circuit resistance only slightly. Thus, the total current drawn from the battery (I') remains approximately the same as the initial current (I<0xE1><0xB5><0x8D>).
     Under this condition (R >> G), when the deflection becomes half (θ/2), the current through the galvanometer (I<0xE1><0xB5><0x8D>') must also be half of the original current (I<0xE1><0xB5><0x8D>/2).
     For I<0xE1><0xB5><0x8D>' = I<0xE1><0xB5><0x8D> / 2, the current must divide equally between G and S when the shunt is connected (assuming total current I' ≈ I<0xE1><0xB5><0x8D>). This happens only if the resistance of the shunt path (S) is equal to the resistance of the galvanometer path (G).
     Therefore, under the condition R >> G, G ≈ S.

• Circuit Diagram (for G):

  graph LR
      A[Battery E] --- K1[Key K1] --- R[Resistance Box R] --- P1 --- G((Galvanometer G)) --- P2 --- A;
      P1 --- K2[Key K2] --- S[Shunt Box S] --- P2;
  

  Initially, only K₁ is closed. Then K₂ is also closed.

(b) Figure of Merit (k):

• Definition: The figure of merit of a galvanometer is defined as the current required to produce a deflection of one division on its scale.
• Calculation: Using the initial setup for the half-deflection method (only R, G, E in series, K₁ closed, K₂ open), let the deflection be θ for a resistance R.
  The current flowing through the galvanometer is I<0xE1><0xB5><0x8D> = E / (R + G).
  This current I<0xE1><0xB5><0x8D> produces a deflection θ.
  Therefore, the current required for one division deflection (figure of merit, k) is:
  k = I<0xE1><0xB5><0x8D> / θ = E / [(R + G) * θ]
• Unit: Ampere per division (A/div) or microampere per division (µA/div).
• Circuit Diagram (for k):

  graph LR
      A[Battery E] --- K1[Key K1] --- R[Resistance Box R] --- G((Galvanometer G)) --- A;
  

  (Effectively the same circuit as the first step for finding G, without the shunt path)

4. Procedure Summary:

• For G:
  1. Set up the circuit as shown for G. Keep K₂ open.
  2. Close K₁ and adjust R to get a large deflection θ (e.g., 30 divisions). Record R and θ.
  3. Close K₂. Adjust the shunt resistance S until the deflection becomes exactly θ/2. Record S.
  4. The resistance of the galvanometer G ≈ S.
  5. Repeat for different values of θ.
• For k:
  1. Use the circuit without the shunt (K₂ open).
  2. Measure the EMF (E) of the battery using a voltmeter or use the known value.
  3. Close K₁ and adjust R to get a measurable deflection θ (e.g., 20-30 divisions). Record R and θ.
  4. Use the value of G calculated previously.
  5. Calculate k using the formula k = E / [(R + G) * θ].
  6. Repeat for different values of R and θ.

5. Key Formulas:

• Resistance of Galvanometer: G ≈ S (Condition: R >> G)
• Figure of Merit: k = E / [(R + G) * θ]

6. Precautions:

• All connections must be neat, clean, and tight. Use sandpaper to clean the ends of connecting wires.
• Use a high resistance box for R so that the condition R >> G is satisfied. This ensures a large initial resistance, protecting the galvanometer from excessive current.
• Always start with the maximum resistance in the box R and then gradually decrease it.
• Insert the key K₁ only when readings are to be taken to avoid unnecessary current drain from the battery.
• The EMF of the battery should remain constant throughout the experiment.
• The shunt resistance S should be low.
• Ensure the galvanometer coil oscillates freely and comes to rest quickly.
• Avoid parallax error while reading the galvanometer deflection.

7. Potential Sources of Error:

• The condition R >> G may not be perfectly satisfied.
• The EMF of the battery might change during the experiment.
• Resistances of the connecting wires are assumed negligible.
• Resistance coils in the boxes may not have their marked values exactly.
• Parallax error in reading the deflection.
• Non-uniform divisions on the galvanometer scale.

Multiple Choice Questions (MCQs):

1. In the half-deflection method for determining galvanometer resistance, the shunt resistance (S) is connected:
   a) In series with the galvanometer
   b) In parallel with the galvanometer
   c) In series with the high resistance R
   d) In parallel with the battery

2. The condition under which the galvanometer resistance G is approximately equal to the shunt resistance S in the half-deflection method is:
   a) R << G
   b) R = G
   c) R >> G
   d) R = S

3. What is the primary purpose of the high resistance R connected in series with the galvanometer initially?
   a) To increase the deflection
   b) To decrease the sensitivity
   c) To protect the galvanometer from high current
   d) To measure the EMF accurately

4. The figure of merit (k) of a galvanometer is defined as:
   a) The resistance required for unit deflection
   b) The voltage required for unit deflection
   c) The current required for unit deflection
   d) The total deflection produced by 1 Ampere

5. The unit of the figure of merit (k) of a galvanometer is:
   a) Ohm (Ω)
   b) Volt per division (V/div)
   c) Ampere per division (A/div)
   d) Ohm per division (Ω/div)

6. In the formula k = E / [(R + G) * θ], what does θ represent?
   a) The half-deflection value
   b) The full deflection obtained with resistance R in series
   c) The angle of the magnetic field
   d) The total number of divisions on the scale

7. Why should the key be inserted only while taking readings?
   a) To prevent sparking
   b) To allow the galvanometer to cool down
   c) To conserve the energy of the battery and maintain constant EMF
   d) To increase the accuracy of resistance R

8. If the EMF of the battery is 2V, the high resistance R is 5000 Ω, the galvanometer resistance G is 100 Ω, and the deflection θ is 20 divisions, what is the figure of merit (k)?
   a) 2.0 x 10⁻⁵ A/div
   b) 1.96 x 10⁻⁵ A/div
   c) 3.92 x 10⁻⁵ A/div
   d) 4.0 x 10⁻⁴ A/div

9. In the half-deflection method, if the initial deflection is 24 divisions, the shunt S is adjusted until the deflection becomes:
   a) 24 divisions
   b) 0 divisions
   c) 12 divisions
   d) 48 divisions

10. Which of the following is a necessary precaution for this experiment?
    a) Using a low resistance box for R
    b) Keeping the shunt resistance S very high
    c) Ensuring the battery EMF is variable
    d) Cleaning the ends of connecting wires with sandpaper

Answers to MCQs:

1. b) In parallel with the galvanometer
2. c) R >> G
3. c) To protect the galvanometer from high current
4. c) The current required for unit deflection
5. c) Ampere per division (A/div)
6. b) The full deflection obtained with resistance R in series (in the context of calculating k)
7. c) To conserve the energy of the battery and maintain constant EMF
8. b) 1.96 x 10⁻⁵ A/div (Calculation: k = 2 / [(5000 + 100) * 20] = 2 / (5100 * 20) = 2 / 102000 ≈ 1.96 x 10⁻⁵ A/div)
9. c) 12 divisions
10. d) Cleaning the ends of connecting wires with sandpaper

Study these notes carefully, focusing on the underlying principles, formulas, and precautions. Understanding the 'why' behind each step is crucial for tackling application-based questions in your exams. Good luck!