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Electricity and Magnetism
Moving Coil Galvanometer
A moving coil galvanometer is an electromechanical measuring instrument used for detecting and indicating small electric currents. It operates on the principle that a current-carrying coil placed in a uniform magnetic field experiences a deflecting torque.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic construction, working principle, conversion of galvanometer into ammeter and voltmeter using shunts and multipliers.
PGT extension
Ballistic galvanometer equations, charge sensitivity, damping, logarithmic decrement, and correction for damping in flux measurements.
Prerequisites
Magnetic effect of electric current and Lorentz force
Torque on a current loop in a magnetic field
Hooke's law of elasticity and torsional constant of a suspension wire
Principle of Moving Coil Galvanometer
A current-carrying coil suspended in a uniform magnetic field experiences a magnetic torque.
The deflecting torque is directly proportional to the current flowing through the coil.
Restoring torque is provided by the elastic twist of the suspension wire or a hairspring.
Radial Magnetic Field
A radial magnetic field is achieved by using a cylindrical soft iron core and concave pole pieces.
In a radial field, the magnetic field lines are always parallel to the plane of the coil for any orientation.
This ensures that the deflecting torque remains independent of the angle of deflection, making the scale linear.
Conversion into Ammeter
A galvanometer is converted into an ammeter by connecting a very low resistance called a shunt in parallel with it.
The shunt bypasses the major portion of the main current, protecting the delicate galvanometer coil.
The effective resistance of the resulting ammeter is extremely small compared to the galvanometer resistance.
Conversion into Voltmeter
A galvanometer is converted into a voltmeter by connecting a very high resistance in series with it.
The high series resistance limits the current passing through the galvanometer when large potential differences are applied.
The effective resistance of the resulting voltmeter is very large, drawing negligible current from the circuit being measured.
Derivations
Torque and Deflection in Galvanometer
Conditions: Radial magnetic field, N turns of coil with area A
Let a coil of N turns and area A carry current I in a radial magnetic field B.
The magnetic force on each vertical side of length l is F = NIbB, where b is the breadth and $A = l * b$.
The deflecting torque is given by ta$u_d = N I A B$.
Equating deflecting torque to restoring torque ta$u_r = k * \theta$, where k is the torsional constant and theta is deflection: N I A $B = k * \theta$.
Result: thet$a = (NAB / k) * I$
Shunt Resistance for Ammeter Conversion
Conditions: Galvanometer resistance G, full-scale deflection current I_g, maximum range I
Let the maximum current to be measured be I, and the current passing through the galvanometer at full-scale deflection be I_g.
The remaining current (I - I_g) must pass through the parallel shunt resistance S.
Since the galvanometer and shunt are in parallel, the potential difference across them is equal: I_g * $G = (I - I_g) * S$.
Solving for S yields the required shunt resistance.
Result: $S = \frac{I_g G}{I - I_g}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Deflection Torque
$$\tau = NIAB$$
N = number of turns, I = current, A = area, B = magnetic field Newton-meter (N*m)
Radial magnetic field
Current Sensitivity
$$S_i = \frac{\theta}{I} = \frac{NAB}{k}$$
theta = deflection, I = current, k = torsional constant radian per ampere (rad/A) or scale divisions per ampere
V = potential difference, G = galvanometer resistance radian per volt (rad/V)
Constant resistance G
Shunt Resistance for Ammeter
$$S = \frac{I_g G}{I - I_g}$$
$I_g = full scale current$, G = galvanometer resistance, I = total current range Ohm (ohm)
Parallel connection
Series Resistance for Voltmeter
$$R = \frac{V}{I_g} - G$$
V = maximum voltage range, $I_g = full scale current$, G = galvanometer resistance Ohm (ohm)
Series connection
Figure of Merit
$$k' = \frac{I}{\theta} = \frac{k}{NAB}$$
I = current, theta = deflection in divisions Ampere per division (A/div)
Single scale division deflection
Ballistic Galvanometer Charge Sensitivity
$$q = \frac{T k}{\pi NAB} \theta_0$$
T = time period of oscillation, q = total charge, thet$a_0 = first maximum deflection$ Coulomb (C)
Transient discharge through ballistic galvanometer
Effective Resistance of Converted Ammeter
$$R_{eff} = \frac{GS}{G + S}$$
G = galvanometer resistance, S = shunt resistance Ohm (ohm)
Parallel combination
Graphs and experiments
Current vs Deflection Graph
Draw: A straight line passing through the origin with deflection theta on the y-axis and current I on the x-axis.
Interpret: The slope of the line represents the current sensitivity (NAB / k) of the galvanometer, confirming linearity due to the radial field.
Exam use: Used to determine current sensitivity and figure of merit experimentally.
Half-Deflection Method for Galvanometer Resistance (G)
Draw: Circuit diagram with a high resistance box in series, galvanometer, battery, and a shunt box across the galvanometer.
Interpret: When the shunt resistance is adjusted such that deflection drops to exactly half, the galvanometer resistance equals the shunt resistance under specific conditions.
Exam use: Standard viva and numerical problem setup for finding internal resistance of a galvanometer.
Ballistic Damped Oscillation Decay
Draw: Logarithmic decay curve of angular amplitude versus time for an open or closed circuit ballistic galvanometer.
Interpret: Used to calculate logarithmic decrement lambda and correction for damping in charge measurements.
Exam use: Advanced PGT level conceptual testing on ballistic damping.
Solved numericals
TGT core
A galvanometer has a resistance of 50 ohm and shows full scale deflection for a current of 2 mA. Calculate the value of the shunt resistance required to convert it into an ammeter of range 0 to 5 A.
Identify given values: G = 50 ohm, $I_g = 2 mA = 2 * 10^-3 A$, I = 5 A.
Apply the shunt formula: $S = (I_g * G) / (I - I_g)$.
A galvanometer has a coil resistance of 30 ohm and gives full-scale deflection for 15 mA. Find the series resistance needed to convert it into a voltmeter reading up to 150 V.
Identify given values: G = 30 ohm, $I_g = 15 mA = 15 * 10^-3 A$, V = 150 V.
Apply the multiplier formula: $R = (V / I_g) - G$.
A moving coil galvanometer has 100 turns, each of area 2 cm^2. The magnetic field is 0.2 T, and the torsional constant of the suspension strip is 10^-6 N*m/rad. Find its current sensitivity.
Identify given parameters: N = 100, $A = 2 cm^2 = 2 * 10^-4 m^2$, B = 0.2 T, $k = 10^-6 N*m/rad$.
Formula for current sensitivity: $S_i = (N * A * B) / k$.
The magnetic field in the gap is assumed to be strictly radial to maintain a linear scale, which holds true only for carefully shaped pole pieces and small deflections.
The restoring torque is assumed to follow Hooke's torsional law linearly, which fails if the elastic limit of the suspension wire is exceeded.
Common misconceptions
Common error: An ideal galvanometer has zero resistance.
Correct: A practical galvanometer has a definite finite coil resistance (G).
The coil is made of fine insulated copper wire which inherently possesses electrical resistance.
Common error: Connecting a shunt increases the current measuring capacity by blocking all current from the galvanometer.
Correct: The galvanometer still carries its full-scale current I_g while the shunt carries the excess current (I - I_g).
Current divides inversely proportional to resistances in a parallel network.
Quick revision
Radial magnetic field ensures linear deflection scale: theta is proportional to I.
Ammeter uses a small parallel shunt; Voltmeter uses a large series multiplier.
Deflection Torque: tau = NIAB
Must remember
The deflecting torque in a moving coil galvanometer is given by $\tau = NIAB$, where $N$ is number of turns, $I$ is current, $A$ is area, and $B$ is the magnetic field.
A radial magnetic field ensures that the deflecting torque remains independent of the angle of deflection, making the scale linear.
A galvanometer is converted into an ammeter by connecting a very low resistance (shunt $S$) in parallel.
A galvanometer is converted into a voltmeter by connecting a very high resistance ($R$) in series.
Current sensitivity is given by $S_i = \frac{\theta}{I} = \frac{NAB}{k}$, and voltage sensitivity by $S_v = \frac{S_i}{G} = \frac{NAB}{kG}.
For a ballistic galvanometer, the charge sensitivity relates to the first maximum deflection $\theta_0$ via $q = \frac{T k}{\pi NAB} \theta_0$.
Glossary
Radial Magnetic FieldA magnetic field configuration where field lines are always parallel to the plane of the coil, achieved using a soft iron core and concave pole pieces.
Shunt Resistance ($S$)A very low resistance connected in parallel with a galvanometer to convert it into an ammeter.
Multiplier ($R$)A very high resistance connected in series with a galvanometer to convert it into an voltmeter.
Figure of Merit ($k'$)The current required to produce a deflection of one scale division in the galvanometer, $k' = \frac{I}{\theta} = \frac{k}{NAB}$.
Torsional Constant ($k$)Restoring torque per unit twist of the suspension wire or hairspring.
Ballistic GalvanometerA specialized galvanometer designed to measure total charge passing through it in a transient discharge.
Ammeter vs. Voltmeter Conversion
Parameter
Ammeter Conversion
Voltmeter Conversion
Connecting Resistor
Low resistance (Shunt, $S$) in parallel
High resistance (Multiplier, $R$) in series
Effective Resistance
Extremely small ($R_{eff} \approx S$)
Very large ($R_{eff} = G + R$)
Ideal Resistance
Zero
Infinite
Purpose
To measure large electric currents
To measure large potential differences
Practice Quiz
Choose an option to reveal the explanation.
1. What is the primary function of a radial magnetic field in a moving coil galvanometer?
Explanation: In a radial magnetic field, the magnetic field lines are always parallel to the plane of the coil for any orientation, making $\tau = NIAB$ constant with respect to angle, thus linearizing the scale.
2. How is a galvanometer converted into an ammeter?
Explanation: Connecting a very low resistance (shunt $S$) in parallel bypasses most of the current, protecting the galvanometer and allowing large currents to be measured.
3. The deflecting torque of a moving coil galvanometer is given by which expression?
Explanation: The magnetic torque experienced by a current-carrying coil with $N$ turns, area $A$, and current $I$ in a magnetic field $B$ is $\tau = NIAB$.
4. What is the formula for the shunt resistance $S$ required to extend the range of a galvanometer of resistance $G$ to a total current $I$, given full-scale deflection current $I_g$?
Explanation: Using potential divider/parallel circuit relations, the potential across the galvanometer $I_g G = (I - I_g)S$, giving $S = \frac{I_g G}{I - I_g}$.
5. Which of the following describes the voltage sensitivity $S_v$ of a galvanometer?
Explanation: Voltage sensitivity is deflection per unit voltage: $S_v = \frac{\theta}{V} = \frac{\theta}{I \cdot G} = \frac{S_i}{G}$.
6. How is a galvanometer converted into a voltmeter?
Explanation: A high series resistance limits the current through the galvanometer when large potential differences are applied, preventing damage.
7. What provides the restoring torque in a moving coil galvanometer?
Explanation: Restoring torque is provided by the elastic twist of the suspension wire or a hairspring, proportional to the angle of twist $\tau_{restoring} = k\theta$.
8. If the number of turns $N$ in a galvanometer coil is doubled, what happens to its current sensitivity $S_i$ (assuming other parameters remain constant)?
Explanation: Current sensitivity $S_i = \frac{NAB}{k}$ is directly proportional to $N$. Doubling $N$ doubles the current sensitivity.
9. What is the effective resistance $R_{eff}$ of an ammeter formed by a galvanometer of resistance $G$ and a shunt $S$?
Explanation: Since the galvanometer and shunt are connected in parallel, their effective resistance is given by the parallel combination formula $\frac{GS}{G + S}$.
10. What is the formula for the series resistance $R$ needed to convert a galvanometer into a voltmeter of range $V$?
Explanation: The total resistance of the voltmeter is $R + G = \frac{V}{I_g}$, therefore $R = \frac{V}{I_g} - G$.
11. For a ballistic galvanometer, what is the charge sensitivity relation with respect to the first maximum deflection $\theta_0$?
Explanation: The charge passing through a ballistic galvanometer is proportional to the first throw: $q = \frac{T k}{\pi NAB} \theta_0$.
12. What is the figure of merit ($k'$) of a galvanometer defined as?
Explanation: Figure of merit is $k' = \frac{I}{\theta}$, which is the current required to produce a deflection of a single scale division.
13. Why is a soft iron core used inside the coil of a moving coil galvanometer?
Explanation: A cylindrical soft iron core along with concave pole pieces creates a strong radial magnetic field, concentrating magnetic field lines.
14. What is a common misconception regarding an ideal galvanometer's resistance?
Explanation: Students often mistakenly think an ideal galvanometer has zero resistance, whereas practical ones have a finite coil resistance $G$.
15. When a shunt is connected across a galvanometer, what happens to the current flowing through the galvanometer coil itself during full-scale deflection?
Explanation: During full-scale deflection, the galvanometer carries its maximum rated current $I_g$, while the shunt carries the excess current $I - I_g$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A galvanometer of resistance $G$ is converted into an ammeter by connecting a shunt resistance $S$ in parallel. If the total current is $I$ and full-scale current is $I_g$, the ratio of the shunt resistance to galvanometer resistance $\frac{S}{G}$ is:
Explanation: From $S(I - I_g) = I_g G$, we get $\frac{S}{G} = \frac{I_g}{I - I_g}$.
2. A moving coil galvanometer has a coil of $N$ turns, area $A$, and is placed in a magnetic field $B$. The torsional constant of the spring is $k$. The current sensitivity of the galvanometer is doubled if:
Explanation: Current sensitivity $S_i = \frac{NAB}{k}$. Doubling $N$ and halving $k$ increases sensitivity by a factor of $2 / (1/2) = 4$, wait. If $N$ is doubled ($2N$) and $k$ is halved ($k/2$), then $S_i' = \frac{(2N)AB}{k/2} = 4 \frac{NAB}{k}$, which quadruples it. Let's check option 1: if $N, A, B$ are doubled, $S_i$ becomes $8$ times. If $N$ is doubled and $k$ is doubled, sensitivity remains unchanged. Let's re-verify option formatting. Actually, if $N$ is doubled ($2N$) and $k$ is doubled ($2k$), sensitivity remains same. If only $k$ is halved, sensitivity is doubled: $S_i' = \frac{NAB}{k/2} = 2 S_i$. Thus, halving $k$ alone doubles sensitivity.
3. A galvanometer has a resistance of $50\ \Omega$ and full-scale deflection current of $2\text{ mA}$. What is the value of the resistance needed to convert it into a voltmeter of range $0 - 10\text{ V}$?
4. If the shunt resistance $S$ in an ammeter conversion is very small compared to the galvanometer resistance $G$, the effective resistance of the ammeter $R_{eff}$ is approximately equal to:
5. In a ballistic galvanometer, the charge sensitivity depends on the time period $T$ of oscillation as:
Explanation: From the formula $q = \frac{T k}{\pi NAB} \theta_0$, charge $q$ is directly proportional to $T$.
6. A galvanometer of resistance $G$ gives full-scale deflection for current $I_g$. A shunt $S_1$ is used to convert it into an ammeter of range $I$, and another shunt $S_2$ is used to convert it into an ammeter of range $2I$. What is the relation between $S_1$ and $S_2$?
Explanation: Using $S = \frac{I_g G}{I_{range} - I_g}$, for range $I$, $S_1 = \frac{I_g G}{I - I_g}$. For range $2I$, $S_2 = \frac{I_g G}{2I - I_g}$. If $I \gg I_g$, $S_1 \approx \frac{I_g G}{I}$ and $S_2 \approx \frac{I_g G}{2I}$, hence $S_1 \approx 2S_2$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What happens to the voltage sensitivity of a galvanometer if its coil resistance $G$ is increased while keeping other factors constant?
Explanation: Voltage sensitivity $S_v = \frac{S_i}{G}$. Since $G$ is in the denominator, increasing $G$ decreases the voltage sensitivity.
2. A galvanometer has a current sensitivity of $10\text{ div/mA}$ and voltage sensitivity of $2\text{ div/V}$. What is the resistance of the galvanometer?
3. Which of the following components helps in creating a radial magnetic field in a moving coil galvanometer?
Explanation: Concave pole pieces and a soft iron cylindrical core shape the magnetic field into a radial pattern.
4. If a galvanometer is wrongly connected in series as an ammeter in a high-current circuit, what is most likely to happen?
Explanation: An ammeter has very low resistance and is connected in parallel (via shunt). Connecting a galvanometer directly in series without a proper shunt exposes it to high current, burning the fine coil.
5. The units of current sensitivity are:
Explanation: Current sensitivity $S_i = \frac{\theta}{I}$, measured in rad/A or scale divisions per ampere.
6. In a ballistic galvanometer, damping is accounted for by measuring:
Explanation: Logarithmic decrement is used in ballistic galvanometer corrections for damping oscillations.
7. Why does a voltmeter need a very high resistance?
Explanation: A high resistance ensures that the voltmeter draws negligible current from the circuit, preserving the circuit conditions.
8. Calculate the shunt resistance required to allow $10\%$ of the main current to pass through a galvanometer of resistance $99\ \Omega$.
Explanation: Here $I_g = 10\% \text{ of } I = 0.1 I$. Shunt $S = \frac{I_g G}{I - I_g} = \frac{0.1 I \times 99}{I - 0.1 I} = \frac{9.9 I}{0.9 I} = 11\ \Omega$. Wait, $9.9 / 0.9 = 11$. Let's check: $I_g G = (I - I_g)S \implies (0.1 I)(99) = (0.9 I) S \implies S = 9.9 / 0.9 = 11\ \Omega$. Correct answer is $11\ \Omega$ (Option B).
9. Which material is commonly used for the suspension wire of a moving coil galvanometer due to its low restoring torque per unit twist and high tensile strength?
Explanation: Phosphor-bronze is preferred for suspension wires because of its small torsional constant and high elasticity/tensile strength.
10. What is the primary difference between a standard moving coil galvanometer and a ballistic galvanometer?
Explanation: A ballistic galvanometer is designed with a large time period so that the entire charge passes through before the coil begins to move significantly.