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Electricity and Magnetism
Applications of Wheatstone Bridge
The Wheatstone bridge is a foundational electrical circuit used for precise measurement of unknown resistances by balancing potential drops across a diamond-shaped bridge network. Its practical applications extend to specialized laboratory instruments and industrial transducers.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Balanced Wheatstone bridge condition, meter bridge calculations, specific resistance determination using wire resistance.
PGT extension
Careful error analysis, sensitivity optimization, Carey Foster bridge for very low resistances, Callendar and Griffith's bridge for platinum resistance thermometers, and Post Office Box networks.
Prerequisites
Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL)
Series and parallel combinations of resistors
Temperature coefficient of resistance
Principle of Balanced Wheatstone Bridge
A bridge consists of four arms containing resistances P, Q, R, and S arranged in a quadrilateral configuration.
A source of electromotive force is connected across one diagonal, and a sensitive galvanometer is connected across the conjugate diagonal.
The bridge is balanced when the galvanometer shows zero deflection, implying that the electric potentials at the two bridge terminals connected to the galvanometer are equal.
Meter Bridge (Slide Wire Bridge)
It is a practical, direct-reading laboratory form of the Wheatstone bridge utilizing a uniform resistance wire of length 1 meter.
Two ratio arms are formed by dividing the wire length at the jockey contact point into lengths l and (100 - l).
It minimizes contact resistance errors by interchanging unknown and known resistances in successive measurements.
Carey Foster Bridge
A differential variant of the Wheatstone bridge specifically designed for the precise comparison of nearly equal low resistances.
It eliminates the end-resistance errors of the bridge board by interchanging the resistances in the left and right outer gaps.
The shift in the null point is directly proportional to the difference between the two low resistances.
Post Office Box
A compact, portable, and robust industrial form of the Wheatstone bridge packaged in a wooden box with plug-type resistance keys.
The ratio arms P and Q are typically set at fixed decimal multiples like 10, 100, or 1000 ohms.
The arm R provides adjustable resistance from 1 to 5000 ohms to achieve the most accurate bridge balance.
Derivations
Condition for Bridge Balance
Conditions: Galvanometer current is zero ($I_g = 0)$
Apply Kirchhoff's Current Law at junction B and D: $I_1 = I_3$ and $I_2 = I_4$.
Apply Kirchhoff's Voltage Law to closed loop ABDA: -I_1 P - I_g G + I_2 R = 0. Since $I_g = 0$, I_1 $P = I_2 R$.
Apply Kirchhoff's Voltage Law to closed loop BCDB: -I_3 Q + I_4 S + I_g G = 0. Since $I_g = 0$, I_3 $Q = I_4 S$.
Substitute $I_3 = I_1$ and $I_4 = I_2 into the second equation: I_1 Q = I_2 S$.
Divide the first equation by the fourth equation: (I_1 P)/(I_1 Q) = (I_2 R)/(I_2 S), which simplifies to P/$Q = R/S$.
Result: P / $Q = R / S$
Sensitivity Optimization of Wheatstone Bridge
Conditions: All four bridge arms have comparable magnitudes of resistance.
Define sensitivity S as the deflection of the galvanometer per unit fractional change in one of the bridge resistances.
Express the galvanometer current as a function of the bridge arms and source voltage using Thevenin's theorem.
Differentiate the galvanometer current expression with respect to the resistance of the unknown arm and equate the derivative to zero.
Show that maximum sensitivity is achieved when all four arm resistances are equal to each other, i.e., P = Q = R = S.
Result: Bridge is most sensitive when P = Q = R = S
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Balanced Wheatstone Bridge Condition
$$\frac{P}{Q} = \frac{R}{S}$$
P, Q are ratio arm resistances; R is standard known resistance; S is unknown resistance ohm (\Omega)
Galvanometer current is zero
Meter Bridge Unknown Resistance
$$S = R \left(\frac{100 - l}{l}\right)$$
R = known resistance, l = balancing length from left end, S = unknown resistance ohm (\Omega)
Uniform cross-section of meter bridge wire
Carey Foster Bridge Resistance Difference
$$S - R = \delta l \times \rho$$
S, R = nearly equal resistances, \delta l = shift in null point, \rho = resistance per unit length of bridge wire ohm (\Omega)
Bridge balanced before and after interchanging resistances in outer gaps
Draw: A schematic layout of a meter bridge showing a 1-meter wire AB, resistance box on left, unknown resistance on right, galvanometer, and jockey connected at balance point l.
Interpret: The null point represents the exact geometric division of the wire where the ratio of resistances matches the ratio of lengths, eliminating internal resistance errors of the battery.
Exam use: Tested via numerical questions on finding unknown resistance or shift in balance point upon parallel addition of resistors.
Galvanometer Deflection vs. Jockey Position
Draw: A graph of galvanometer current (I_g) on the y-axis against jockey position (l) on the x-axis, showing a straight line crossing zero at the balance length l_0.
Interpret: The slope of the line indicates the bridge sensitivity; a steeper slope implies higher sensitivity to small position changes near balance.
Exam use: Theoretical understanding of experimental precision and choice of galvanometer.
Platinum Resistance vs. Temperature Calibration Curve
Draw: A linear or slightly curved plot of resistance R_t versus temperature t (^\circ\text{C}) for a platinum wire utilized in a Callendar-Griffiths bridge.
Interpret: Enables direct conversion of measured resistance values into accurate thermodynamic temperature readings.
Exam use: Problems involving temperature coefficients of resistance and thermal bridge calibration.
Solved numericals
TGT core
In a meter bridge, the null point is found at a distance of 40 cm from the left end when a resistance of 12 ohm is connected in the right gap. Find the unknown resistance in the left gap.
Identify the known parameters: right gap resistance S = 12 ohm, balancing length l = 40 cm.
Determine the remaining length of the wire: 100 - $l = 100 - 40 = 60 cm$.
Use the meter bridge formula: P / $Q = l / (100 - l)$, where P is the unknown resistance in the left gap and Q corresponds to the right gap resistance S.
Four resistors of resistance 2 ohm, 2 ohm, 2 ohm, and 3 ohm are connected to form the first three and fourth arms of a Wheatstone bridge. What resistance must be connected in parallel with the 3 ohm resistor to balance the bridge?
Let the arms be P = 2 ohm, Q = 2 ohm, R = 2 ohm, and the fourth arm has S' = 3 ohm initially.
For a balanced bridge, the condition is P / $Q = R / S_eq$.
Substitute the values: 2 / $2 = 2 / S_eq$, which gives $S_eq = 2 ohm$.
Let the required parallel resistance be x. The equivalent resistance of 3 ohm and x in parallel is given by (3 * x) / (3 + x) = 2.
Solve for x: 3$x = 2(3 + x) => 3x = 6 + 2x => x = 6 ohm$.
Answer: 6 \Omega
PGT extension
In a Carey Foster bridge, two nearly equal resistances P and Q are placed in the inner gaps. With a standard resistance of 1.0 ohm in the left outer gap and an unknown resistance in the right outer gap, the null point is obtained at 45 cm. When the resistances in the outer gaps are interchanged, the null point shifts by 2 cm. If the resistance of the bridge wire is 0.1 ohm per cm, calculate the value of the unknown resistance.
Recall the Carey Foster bridge relation for resistance difference: S - $R = \delta l \times \rho$, where \delta l is the shift in null point and \rho is resistance per unit length.
Identify the given values: \delta l = 2 cm, $\rho = 0.1 \Omega/\text{cm}$, and the known resistance R = 1.0 \0hm.
Calculate the difference: S - R = 2 \times 0.1 = 0.2 \Omega.
Substitute $R = 1.0 \Omega into the equation: S - 1.0 = 0.2$.
Solve for the unknown resistance S: $S = 1.0 + 0.2 = 1.2 \Omega$.
Answer: 1.2 \Omega
Assumptions and limitations
Connecting wire and terminal resistances (end errors) are neglected unless specialized calibration bridges like Carey Foster are used.
Heating effects due to current flow through the bridge arms are assumed negligible so that resistances remain constant during measurement.
Common misconceptions
Common error: Interchanging the battery and galvanometer terminals alters the balanced condition of a Wheatstone bridge.
Correct: Interchanging the battery and galvanometer leaves the balance condition P/$Q = R/S entirely unchanged$.
The bridge balance symmetry relies strictly on the ratio of adjacent and opposite arm resistances, which remains invariant under interchange of conjugate diagonals.
Common error: A Wheatstone bridge can accurately measure extremely low resistances without special modifications.
Correct: Standard Wheatstone bridges cannot measure very low resistances accurately due to the comparable magnitude of lead and contact resistances.
Contact resistances become a significant fraction of very low unknown resistances, necessitating Kelvin's double bridge instead.
Quick revision
Balanced Wheatstone bridge condition: P / $Q = R / S with zero galvanometer current$.
Meter Bridge Unknown Resistance: $S = R \left(\frac{100 - l}{l}\right)$
Must remember
The balanced condition of a Wheatstone bridge is given by $\frac{P}{Q} = \frac{R}{S}$.
Interchanging the battery and galvanometer terminals does not affect the balance condition of the bridge.
A Meter Bridge utilizes a 1 meter long uniform wire to find unknown resistances using the formula $S = R \left(\frac{100 - l}{l}\right)$.
Carey Foster Bridge is a differential variant of the Wheatstone bridge designed for the precise comparison of nearly equal low resistances.
Post Office Box is a compact industrial form of Wheatstone bridge with fixed ratio arms and a variable rheostat arm.
Glossary
Wheatstone BridgeAn electrical network of four resistances used to measure an unknown resistance by balancing potentials across a galvanometer.
Meter BridgeA practical form of Wheatstone bridge utilizing a 1-meter long uniform resistance wire.
Carey Foster BridgeA differential Wheatstone bridge used for precise measurement of small resistance differences.
Post Office BoxA portable plug-type industrial realization of the Wheatstone bridge containing ratio and rheostat arms.
Comparison of Wheatstone Bridge Practical Forms
Feature
Meter Bridge
Carey Foster Bridge
Post Office Box
Primary Purpose
General resistance and resistivity measurement
Precise comparison of nearly equal low resistances
Portable industrial measurement of unknown resistances
Bridge Wire Length
1 meter
Usually 1 meter or more
No bridge wire; plug resistance box used
Error Elimination
Interchanging known and unknown resistances
Interchanging resistances in left and right outer gaps
Using standardized plug-type precision resistors
Practice Quiz
Choose an option to reveal the explanation.
1. What is the condition for a Wheatstone bridge to be balanced?
Explanation: A Wheatstone bridge is balanced when the galvanometer shows zero deflection, which yields the condition $\frac{P}{Q} = \frac{R}{S}$.
2. What happens to the balance condition of a Wheatstone bridge if the battery and galvanometer terminals are interchanged?
Explanation: The balance symmetry of the bridge relies strictly on adjacent and opposite arm resistance ratios, which remain invariant under the interchange of conjugate diagonals.
3. In a meter bridge, if the balancing length from the left end is $l$, what is the expression for the unknown resistance $S$ in terms of known resistance $R$?
Explanation: The ratio arms correspond to lengths $l$ and $(100 - l)$, leading to the formula $S = R \left(\frac{100 - l}{l}\right)$.
4. Which bridge is specifically designed for the precise comparison of nearly equal low resistances?
Explanation: The Carey Foster bridge is a differential variant of the Wheatstone bridge used for precise comparison of nearly equal low resistances and eliminating end-resistance errors.
5. In a Post Office Box, what are the typical values set for the fixed ratio arms $P$ and $Q$?
Explanation: The ratio arms $P$ and $Q$ in a Post Office Box are typically set at fixed decimal multiples like $10\ \Omega$, $100\ \Omega$, or $1000\ \Omega$.
6. Why are standard Wheatstone bridges unsuitable for measuring very low resistances directly?
Explanation: Contact and lead resistances introduce significant errors when measuring very low resistances, which necessitates specialized configurations like Kelvin's double bridge.
7. What is the formula for the specific resistance $\rho$ obtained using a meter bridge for a wire of diameter $d$, length $L$, and measured resistance $S$?
Explanation: Using $S = \rho \frac{L}{A}$ where cross-sectional area $A = \frac{\pi d^2}{4}$, we get $\rho = \frac{S \pi d^2}{4 L}$.
8. In the Carey Foster bridge formula $S - R = \delta l \times \rho$, what does $\rho$ represent?
Explanation: In the Carey Foster bridge equation, $\rho$ represents the resistance per unit length of the bridge wire.
9. How is temperature determined using a platinum resistance thermometer based on a Wheatstone bridge?
Explanation: Assuming a linear temperature dependence, temperature $t$ is calculated using $t = \frac{R_t - R_0}{R_{100} - R_0} \times 100$.
10. What is the industrial form of a Wheatstone bridge packaged in a wooden box with plug-type resistance keys called?
Explanation: A Post Office Box is a compact, portable, and robust industrial form of the Wheatstone bridge packaged with plug-type resistance keys.
11. If the ratio arms $P$ and $Q$ of a Post Office Box are $10\ \Omega$ and $1000\ \Omega$ respectively, and the rheostat arm $R$ is $24\ \Omega$, what is the unknown resistance $S$?
Explanation: Using $S = R \times \left(\frac{Q}{P}\right) = 24 \times \left(\frac{1000}{10}\right) = 24 \times 100 = 2400\ \Omega$.
12. What is the role of the galvanometer in a Wheatstone bridge circuit?
Explanation: A sensitive galvanometer is connected across the conjugate diagonal to detect zero current, indicating that the bridge is balanced.
13. Which of the following parameters maximizes the sensitivity of a Wheatstone bridge?
Explanation: A Wheatstone bridge is most sensitive when all four arms have resistances of the same order of magnitude.
14. What is the Open-circuit Thevenin equivalent voltage $V_{th}$ across the galvanometer terminals of an unbalanced Wheatstone bridge with source EMF $E$ and arm resistances $P, Q, R, S$?
Explanation: The Thevenin voltage across the galvanometer branch is given by $V_{th} = E \left(\frac{R}{P+R} - \frac{S}{Q+S}\right)$.
15. In a meter bridge experiment, end errors are minimized by:
Explanation: Interchanging unknown and known resistances in successive measurements cancels out end-resistance errors caused by copper strips at the ends of the meter bridge.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. In a meter bridge, the null point is obtained at $40\ \text{cm}$ from the left end when a known resistance $R$ is in the left gap and an unknown resistance $S$ is in the right gap. If $R = 12\ \Omega$, what is the value of $S$?
2. A Carey Foster bridge has a bridge wire of resistance per unit length $\rho = 0.1\ \Omega/\text{cm}$. When two nearly equal resistances $S$ and $R$ are compared, the shift in null point is found to be $5\ \text{cm}$. What is the difference between $S$ and $R$?
Explanation: Using $S - R = \delta l \times \rho = 5\ \text{cm} \times 0.1\ \Omega/\text{cm} = 0.5\ \Omega$.
3. In a Wheatstone bridge, the resistances of arms $P, Q, R, S$ are $10\ \Omega, 20\ \Omega, 30\ \Omega$, and $60\ \Omega$ respectively. Is the bridge balanced, and if not, what resistance should be connected in parallel with $S$ to balance it?
Explanation: Check balance condition: $\frac{P}{Q} = \frac{10}{20} = \frac{1}{2}$ and $\frac{R}{S} = \frac{30}{60} = \frac{1}{2}$. Since ratios are equal, the bridge is already balanced.
4. A platinum resistance thermometer has a resistance of $2\ \Omega$ at $0^\circ\text{C}$ and $6\ \Omega$ at $100^\circ\text{C}$. When immersed in a liquid, its resistance becomes $5\ \Omega$. What is the temperature of the liquid?
5. In a Post Office Box, the ratio arms are set at $P = 10\ \Omega$ and $Q = 100\ \Omega$. To balance the bridge, a resistance of $345\ \Omega$ is removed from the rheostat arm $R$. What is the unknown resistance $S$?
Explanation: Using $S = R \times \left(\frac{Q}{P}\right) = 345 \times \left(\frac{100}{10}\right) = 345 \times 10 = 3450\ \Omega$.
6. If a meter bridge wire of length $1\ \text{m}$ has a uniform cross-section and total resistance of $10\ \Omega$, what is the resistance of the segment corresponding to balancing length $l = 30\ \text{cm}$?
Explanation: Since resistance is uniformly distributed along length, resistance for $30\ \text{cm}$ out of $100\ \text{cm}$ with total resistance $10\ \Omega$ is $10 \times \frac{30}{100} = 3\ \Omega$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Which of the following bridge circuits is best suited for measuring high resistances?
Explanation: Meggers or high-resistance modification setups are used for high resistances, whereas Kelvin bridge is for very low and Wheatstone for moderate resistances.
2. In a meter bridge, thermal EMFs generated at junctions can cause errors. How are these errors minimized in practice?
Explanation: Reversing the battery current reverses the direction of the bridge current while thermal EMFs remain constant, allowing their effect to be canceled by taking the mean of two sets of readings.
3. What is the main advantage of using a Carey Foster bridge over a standard meter bridge?
Explanation: Carey Foster bridge eliminates end-resistance errors by interchanging resistances in the left and right outer gaps, making it extremely precise for small resistance differences.
4. In a balanced Wheatstone bridge, if all four resistances $P, Q, R, S$ are doubled, what happens to the balance condition?
Explanation: Since the balance condition depends on the ratio $\frac{P}{Q} = \frac{R}{S}$, doubling all resistances preserves the ratio, so the bridge remains balanced.
5. Why is the bridge wire in a meter bridge made of an alloy like Manganin or Constantan instead of Copper?
Explanation: Alloys like Manganin and Constantan have high resistivity and very low temperature coefficients of resistance, ensuring resistance doesn't change significantly with Joule heating during measurements.
6. Calculate the current $I_g$ through the galvanometer in an unbalanced bridge if $V_{th} = 0.5\ \text{V}$, $R_{th} = 150\ \Omega$, and galvanometer resistance $G = 100\ \Omega$.
7. In a Post Office Box, why are plug keys made of solid brass fitted into tapered brass blocks?
Explanation: Solid brass blocks and tapered keys minimize contact resistance and provide a firm, reliable electrical connection.
8. If the length of a meter bridge wire is increased while keeping its total resistance constant, how does the sensitivity of the bridge change?
Explanation: A longer bridge wire distributes the resistance more gradually per unit length, allowing more precise localization of the null point and increasing sensitivity.
9. In an experiment to determine specific resistance using a meter bridge, the radius of the wire is measured with a screw gauge. If the percentage error in radius measurement is $2\%$, what is the resulting percentage error in calculated resistivity $\rho$ (assuming length and resistance measurements have negligible error)?
Explanation: Since $\rho = \frac{S \pi d^2}{4 L} \propto d^2$, the relative error in $\rho$ due to radius/diameter is $2 \times (\text{percentage error in } d) = 2 \times 2\% = 4\%$.
10. What is the primary reason for tapping the key of the galvanometer gently rather than keeping it continuously closed while searching for a null point?
Explanation: Tapping protects the sensitive galvanometer from large currents when far from balance and prevents thermal changes due to sustained current flow.