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Electricity and Magnetism
Voltmeter
A voltmeter is a high-resistance measuring instrument used to determine the potential difference between two points in an electrical circuit. It is always connected in parallel across the components whose potential drop is to be measured to minimize current diversion.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Conversion of a galvanometer into a voltmeter by connecting a high series resistance, calculation of multiplier resistance, and basic circuit connection rules.
PGT extension
Loading effect of voltmeters, finite resistance limitations, multi-range voltmeter design, sensitivity expressed in ohms per volt, and calibration using potentiometer arrangements.
Prerequisites
Ohm's Law and circuit potential distribution
Moving coil galvanometer principles and torque equation
Series and parallel resistance combinations
Basic Principle of Conversion
A moving coil galvanometer cannot be directly connected across a high potential difference due to its low coil resistance and small full-scale deflection current.
To convert a galvanometer into a voltmeter, a very high resistance (multiplier, R) is connected in series with the galvanometer coil.
The high series resistance limits the total current flowing through the device, protecting the delicate coil from thermal damage while scaling the voltage range.
Ideal vs. Practical Voltmeter
An ideal voltmeter possesses infinite internal resistance so that it draws zero current from the circuit under test and does not disturb the operating voltages.
A practical voltmeter has a very high, but finite, internal resistance which draws a minute fraction of circuit current, causing a slight loading effect.
Higher resistance per volt (sensitivity) characterizes a better quality voltmeter with minimal circuit disturbance.
Voltmeter Sensitivity
Voltmeter sensitivity (S) is defined as the reciprocal of the full-scale deflection current (I_g).
It is expressed in ohms per volt (Ω/V), signifying how much resistance the voltmeter introduces per volt of its full-scale rating.
Sensitivity helps determine the loading error when measuring voltages across high-impedance circuit nodes.
Multi-range Voltmeter Design
A multi-range voltmeter incorporates multiple series resistors connected via a selector switch to provide different voltage measurement scales.
Resistors are calculated individually for each desired voltage range using the galvanometer constant parameters.
Care must be taken to ensure that switching between ranges maintains continuous circuit protection without open-circuiting the coil.
Derivations
Derivation of Series Resistance (Multiplier) for Voltmeter Conversion
Conditions: Galvanometer of resistance G with full-scale deflection current I_g to be converted to measure a maximum potential difference V.
Let the total resistance of the voltmeter be $R_v = G + R$, where R is the series resistance to be added.
According to Ohm's law, the maximum voltage V that can be measured corresponds to the full-scale deflection current I_g passing through the total resistance R_v.
Write the equation: $V = I_g (G + R)$.
Rearrange the equation to solve for the required series resistance R: $R = (V / I_g) - G$.
Result: $R = \frac{V}{I_g} - G$
Derivation of Voltmeter Sensitivity and Range Extension
Conditions: Extending an existing voltmeter range or defining sensitivity S in terms of full-scale current I_g.
Define sensitivity S as $S = \frac{1}{I_g}$.
Substitute $I_g = \frac{1}{S} into the multiplier formula: R = V \cdot S - G$.
If a multiplier n is defined as the factor by which the voltage range is increased ($n = V / V_g)$, then $V = n I_g G$.
Substitute V into the resistance formula to obtain $R = G(n - 1)$.
R = series resistance, V = maximum voltage to be measured, $I_g = galvanometer full-scale current$, G = galvanometer resistance R in ohms (Ω), V in volts (V), I_g in amperes (A), G in ohms (Ω)
Valid for DC moving coil galvanometer conversion
Total Resistance of Voltmeter
$$R_v = G + R$$
$R_v = total voltmeter resistance$, G = galvanometer resistance, R = series resistance Ohms (Ω)
Series combination
Voltmeter Sensitivity
$$S = \frac{1}{I_g}$$
S = sensitivity, $I_g = full-scale deflection current$ Ohms per volt (Ω/V)
Constant magnetic field and linear coil response
Range Multiplication Factor
$$n = \frac{V}{V_g} = \frac{V}{I_g G}$$
n = multiplication factor, V = new voltage range, $V_g = initial voltage range$ Dimensionless
Linear scale instruments
Multiplier Resistance via Multiplication Factor
$$R = G(n - 1)$$
R = series resistance, G = galvanometer resistance, n = multiplication factor Ohms (Ω)
Non-ideal voltmeter connected across a high source resistance
Power Dissipation in Voltmeter
$$P = \frac{V^2}{R_v}$$
P = power consumed, V = applied voltage, $R_v = voltmeter internal resistance$ Watts (W)
Steady-state DC conditions
Current Drawn by Voltmeter
$$I = \frac{V}{R_v}$$
I = current drawn, V = measured voltage, $R_v = voltmeter resistance$ Amperes (A)
Ohmic resistance
Graphs and experiments
Calibration of Voltmeter using Potentiometer
Draw: A circuit diagram showing a DC potentiometer wire connected in series with a driver cell and rheostat, a standard cell for standardization, and the test voltmeter connected across a variable potential divider tap along with a standard potentiometer null detector.
Interpret: Compares the reading indicated by the voltmeter against the exact electromotive force/potential difference measured accurately by the potentiometer without drawing current at null point.
Exam use: Tested in practical physics viva and numerical problems regarding systematic error correction in voltmeters.
Voltmeter Loading Error Graph
Draw: A plot of measured voltage ratio (V_m / V_t) on the y-axis against the ratio of source resistance to voltmeter resistance (R_{th} / R_v) on the x-axis, showing a hyperbolic decay from 1.
Interpret: Illustrates that as the internal resistance of the voltmeter becomes comparable to the source resistance, the measured voltage drops significantly below the true open-circuit voltage.
Exam use: Conceptual PGT multiple-choice questions on measurement accuracy and finite resistance effects.
Linear Calibration Curve of a Voltmeter
Draw: A straight-line graph plotting Voltmeter Scale Reading (y-axis) versus Potentiometer True Voltage (x-axis) passing through the origin with a slope equal to unity for an ideal instrument.
Interpret: Any deviation from the straight-line slope indicates calibration errors, zero errors, or non-linearity in the spring restoring torque of the galvanometer.
Exam use: Used to evaluate experimental observation tables and graphical error analysis questions.
Solved numericals
TGT core
A galvanometer has a resistance of 50 Ω and gives a full-scale deflection for a current of 2 mA. Calculate the value of the series resistance required to convert it into a voltmeter of range 0 to 10 V.
Identify given values: G = 50 \Omega, $I_g = 2 \text{ mA} = 2 \times 10^{-3} \text{ A}$, V = 10 \text{ V}.
Apply the formula for series resistance: $R = \frac{V}{I_g} - G$.
Substitute the values: $R = \frac{10}{2 \times 10^{-3}} - 50$.
Calculate: $R = 5000 - 50 = 4950 \Omega$.
Answer: 4950 \Omega
PGT extension
A voltmeter with a resistance of 10 kΩ is used to measure the potential difference across a 20 kΩ resistor connected in series with a 50 kΩ resistor across a 140 V DC supply. Find the percentage error in the voltage reading caused by the loading effect of the voltmeter.
Calculate the true potential difference across the 20 kΩ resistor without the voltmeter: $V_{true} = 140 \times \frac{20}{20 + 50} = 140 \times \frac{2}{7} = 40 \text{ V}$.
When the voltmeter of $R_v = 10 \text{ k}\Omega$ is connected in parallel with the 20 kΩ resistor, calculate the equivalent parallel resistance R_p: $R_p = \frac{20 \times 10}{20 + 10} = \frac{200}{30} = 6.67 \text{ k}\Omega$.
Calculate the total circuit resistance with the voltmeter connected: $R_{total} = R_p + 50 = 6.67 + 50 = 56.67 \text{ k}\Omega$.
Calculate the total current drawn from the 140 V supply: I = \frac{140}{56.67} \text{ mA}.
Calculate the measured voltage V_{measured} across the parallel combination: $V_{measured} = I \times R_p = \frac{140}{56.67} \times 6.67 = 16.48 \text{ V}$.
Alternatively, using voltage divider with R_p and 50 k\Omega: $V_{measured} = 140 \times \frac{6.67}{6.67 + 50} = 140 \times \frac{6.67}{56.67} = 16.48 \text{ V}$.
Answer: -58.8% (voltmeter reads 16.48 V instead of true 40 V due to severe loading)
PGT extension
A moving coil galvanometer has a resistance of 30 Ω and shows full scale deflection for 50 mA. How can it be converted into a multi-range voltmeter to measure 0-10 V, 0-50 V, and 0-100 V using a single galvanometer?
Given parameters: G = 30 \Omega, $I_g = 50 \text{ mA} = 0.05 \text{ A}$.
For range $V_1 = 10 \text{ V}: R_1 = \frac{10}{0.05} - 30 = 200 - 30 = 170 \Omega$.
For range $V_2 = 50 \text{ V}: Total resistance required in series R_{total2} = \frac{50}{0.05} = 1000 \Omega$. The additional series resistor between 10V and 50V taps is $R_2 = 1000 - 200 = 800 \Omega$.
For range $V_3 = 100 \text{ V}: Total resistance required R_{total3} = \frac{100}{0.05} = 2000 \Omega$. The additional series resistor between 50V and 100V taps is $R_3 = 2000 - 1000 = 1000 \Omega$.
Answer: Series resistors of 170 \Omega, 800 \Omega, and 1000 \Omega connected in a step-ladder arrangement.
Assumptions and limitations
Assumption: Galvanometer deflection is strictly linear with current, which holds only within elastic limits of the control hairsprings.
Limitation: Standard moving coil DC voltmeters cannot directly measure alternating voltages without rectification.
Common misconceptions
Common error: An ideal voltmeter has zero resistance so that it does not impede current flow.
Correct: An ideal voltmeter has infinite resistance.
Infinite resistance ensures that no current is diverted from the main circuit through the voltmeter branch.
Common error: Connecting a voltmeter in series with a circuit component measures the voltage across it.
Correct: A voltmeter must always be connected in parallel across the component.
Parallel connection ensures the same potential difference exists across both the component and the voltmeter terminals.
Common error: A voltmeter's range can be increased by connecting a low shunt resistance in parallel.
Correct: A high resistance must be connected in series.
Shunt resistance is used for ammeters to divert current, whereas voltmeters require high series resistance to drop excess voltage.
Quick revision
Voltmete$r = Galvanometer + High Series Resistance$.
Ideal Voltmeter Resistance = Infinity.
Required Series Resistance: $R = (V / I_g) - G$.
Voltmeter Sensitivity $S = 1 / I_g (measured in Ω/V)$.
Always connected in parallel across circuit elements.
Must remember
To convert a moving coil galvanometer into a voltmeter, a very high resistance called a multiplier ($R$) is connected in series with the galvanometer coil.
An ideal voltmeter possesses infinite internal resistance and draws zero current from the circuit under test.
Voltmeter sensitivity ($S$) is defined as the reciprocal of the full-scale deflection current ($I_g$), expressed in ohms per volt ($\omega/V$).
The series resistance required to extend the voltage range up to $V$ is given by $R = \frac{V}{I_g} - G = G(n - 1)$, where $n$ is the range multiplication factor.
A practical voltmeter causes a loading effect due to its finite internal resistance, leading to an apparent measured voltage lower than the true open-circuit voltage.
Glossary
Multiplier ($R$)A very high resistance connected in series with a galvanometer to convert it into a voltmeter.
Voltmeter Sensitivity ($S$)The reciprocal of the galvanometer full-scale deflection current, measured in $Omega/V$.
Loading EffectThe disturbance caused in a circuit when a non-ideal voltmeter draws a minute fraction of current, altering the node voltages.
Range Multiplication Factor ($n$)The ratio of the new maximum voltage range to the initial voltage range of the galvanometer ($n = \frac{V}{V_g}$).
Ideal vs. Practical Voltmeter
Parameter
Ideal Voltmeter
Practical Voltmeter
Internal Resistance
Infinite ($\infty$)
Very high, but finite
Current Drawn
Zero ($0$)
Minute fraction of circuit current
Circuit Disturbance
None
Slight loading effect
Voltmeter vs. Ammeter Conversion
Parameter
Voltmeter
Ammeter
Device Modification
High resistance in series
Low shunt resistance in parallel
Connection in Circuit
Parallel
Series
Desired Effective Resistance
As high as possible
As low as possible
Practice Quiz
Choose an option to reveal the explanation.
1. How is a galvanometer converted into a voltmeter?
Explanation: To convert a galvanometer into a voltmeter, a very high resistance (multiplier) is connected in series with the galvanometer coil to limit the total current and drop excess voltage.
2. What is the internal resistance of an ideal voltmeter?
Explanation: An ideal voltmeter has infinite internal resistance so that it draws zero current from the circuit under test and does not disturb operating voltages.
3. Voltmeter sensitivity ($S$) is expressed in which of the following units?
Explanation: Voltmeter sensitivity is defined as $S = \frac{1}{I_g}$, which is expressed in ohms per volt ($\Omega/V$).
4. If a galvanometer of resistance $G$ and full-scale current $I_g$ is to be converted into a voltmeter of range $V$, the series resistance $R$ is given by:
Explanation: Using Ohm's law, total resistance $R_{total} = \frac{V}{I_g} = G + R$, which gives the series resistance multiplier as $R = \frac{V}{I_g} - G$.
5. Why must a voltmeter be connected in parallel across a circuit component?
Explanation: A voltmeter measures potential difference, so a parallel connection ensures that the exact same voltage drops across the voltmeter terminals as across the component.
6. What is the range multiplication factor ($n$) defined as in voltmeter conversion?
Explanation: The multiplication factor is defined as $n = \frac{V}{V_g} = \frac{V}{I_g G}$, representing the ratio of the new voltage range to the initial galvanometer voltage range.
7. What is the effect of connecting a practical voltmeter across a high-impedance circuit node?
Explanation: A practical voltmeter has finite resistance and draws a minute current, forming a voltage divider with the source resistance, leading to a loading error (apparent voltage measured is less than $V_{true}$).
8. If the range multiplication factor $n = 5$ and galvanometer resistance $G = 50\;\Omega$, what is the required series resistance $R$?
Explanation: Using the formula $R = G(n - 1) = 50 \times (5 - 1) = 50 \times 4 = 200\;\Omega$.
9. How does sensitivity affect the quality of a voltmeter?
Explanation: Higher resistance per volt (sensitivity) means the voltmeter draws less current, introducing minimal circuit disturbance and higher measurement accuracy.
10. What is the expression for the total resistance ($R_v$) of a converted voltmeter?
Explanation: Since the multiplier resistance $R$ is connected in series with the galvanometer resistance $G$, the total resistance of the voltmeter is $R_v = G + R$.
11. Which common misconception about voltmeters is scientifically incorrect?
Explanation: An ideal voltmeter has infinite resistance, not zero resistance. Zero resistance describes an ideal ammeter.
12. What is the formula for the current drawn by a voltmeter when measuring a voltage $V$?
Explanation: By Ohm's law, the current drawn by a voltmeter is the measured voltage divided by the total internal resistance of the voltmeter ($I = \frac{V}{R_v}$).
13. In a multi-range voltmeter design, how are different voltage scales achieved?
Explanation: A multi-range voltmeter uses multiple series resistors connected through a selector switch to provide different voltage measurement scales.
14. What is the power dissipation ($P$) in a voltmeter of resistance $R_v$ connected across a voltage $V$?
Explanation: Power dissipation in an ohmic resistance is given by $P = \frac{V^2}{R_v}$.
15. Why cannot a moving coil galvanometer be connected directly across a high potential difference?
Explanation: A moving coil galvanometer has low coil resistance and a small full-scale deflection current; direct connection across a high potential difference would result in excessive current causing thermal damage.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A galvanometer has a resistance of $50\;\Omega$ and full-scale deflection current of $2\;\text{mA}$. What high resistance $R$ must be connected in series to convert it into a voltmeter of range $0 - 10\;\text{V}$?
3. A voltmeter of internal resistance $10000\;\Omega$ is used to measure potential difference across a circuit with Thevenin resistance $R_{th} = 10000\;\Omega$. If the true open-circuit voltage is $10\;\text{V}$, what voltage will the voltmeter read (loading effect)?
4. If the voltage range of a galvanometer-based voltmeter is to be increased 5 times ($n = 5$), the multiplier resistance $R$ required in terms of galvanometer resistance $G$ is:
Explanation: Using the relation $R = G(n - 1)$, for $n = 5$, we get $R = G(5 - 1) = 4G$.
5. A galvanometer with resistance $G = 20\;\Omega$ shows full-scale deflection for $I_g = 10\;\text{mA}$. The voltage range $V_g$ of the unconverted galvanometer is:
Explanation: Initial voltage range $V_g = I_g \times G = 10 \times 10^{-3} \times 20 = 0.2\;\text{V}$.
6. Which of the following statements regarding voltmeter conversion is correct?
Explanation: Converting a galvanometer into a voltmeter always requires a high series resistance to drop the excess potential difference safely.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. If a voltmeter's sensitivity is $S = 20\;\text{k}\Omega/\text{V}$, what is its full-scale deflection current $I_g$?
2. A galvanometer of resistance $G$ requires series resistance $R_1$ for range $V$ and series resistance $R_2$ for range $2V$. Which relation holds true?
Explanation: For range $V$: $R_1 = \frac{V}{I_g} - G \implies \frac{V}{I_g} = R_1 + G$. For range $2V$: $R_2 = \frac{2V}{I_g} - G = 2\left(\frac{V}{I_g}\right) - G = 2(R_1 + G) - G = 2R_1 + G$.
3. What happens to the total resistance and sensitivity of a voltmeter when a higher series multiplier resistance is added?
Explanation: Total resistance $R_v = G + R$ increases with higher $R$. Sensitivity $S = \frac{1}{I_g}$ depends on galvanometer current, but higher effective ohms-per-volt scaling accompanies higher series resistance for expanded voltage ranges.
4. In designing a multi-range voltmeter, switching between ranges must ensure:
Explanation: Care must be taken when switching ranges to maintain continuous circuit protection without open-circuiting the galvanometer coil, which could cause damage from inductive spikes or surges.
5. Calculate the current drawn by a voltmeter with total resistance $R_v = 20\;\text{k}\Omega$ when connected across a $100\;\text{V}$ supply.
6. Why is the loading effect more pronounced when measuring voltages across high-resistance sources?
Explanation: When source resistance is high, the finite voltmeter resistance forms a voltage divider that drops a noticeable portion of the voltage across the source internal resistance, causing a large loading error.
7. A galvanometer has $G = 100\;\Omega$ and $I_g = 1\;\text{mA}$. To make a voltmeter reading up to $100\;\text{V}$, what is the multiplier resistance?
8. Which of the following factors does NOT affect the conversion of a galvanometer into a voltmeter?
Explanation: The formula $R = \frac{V}{I_g} - G$ depends only on $V$, $I_g$, and $G$. Ambient temperature is not a primary design parameter in basic conversion calculations.
9. What is the power dissipated in a voltmeter having a resistance of $20\;\text{k}\Omega$ when operating at its full-scale voltage of $100\;\text{V}$?