Wheatstone Bridge

Wheatstone bridge is a device that is used to find the resistance of a conductor, in 1842, scientist Wheatstone proposed a theory, which is called the principle of Wheatstone bridge after his name. we can prove or establish the formula for Wheatstone by using Kirchhoff laws. Wheatstone bridge is simply an electric circuit used to measure an unknown electric resistance by balancing two-point of a bridge. Let’s get started!

Wheatstone Bridge Definition

Wheatstone Bridge is an instrument designed to measure unknown resistance in electrical circuits. It calculates the unknown resistance by balancing the two legs of the bridge circuit where one leg contains both known resistors and the other leg contains one known (variable) and one unknown resistor. Since it estimates unknown resistance in an electric circuit, it is also known as a resistance bridge. Wheatstone bridge is a very reliable instrument as it measures the resistance very precisely.

Wheatstone Bridge Principle

Wheatstone Bridge works on the principle of null deflection i.e., there is no current flowing through the galvanometer, and its needle shows no deflection, hence the name null deflection. In the unbalanced state of the Wheatstone bridge i.e., when the potential across the galvanometer is different, the galvanometer shows the deflection, and as the bridge becomes balanced by changing the variable resistor, the potential difference across the galvanometer becomes zero i.e., the equilibrium state of Wheatstone bridge.

Construction of Wheatstone Bridge

Construction of Wheatstone Bridge requires four resistors P, Q, R, and S that are placed in the form of four sides AB, BC, AD, and DC of a quadrilateral ABCD. A cell E and key K₁ are placed between the A and C ends of this quadrilateral, and a sensitive galvanometer G and key K₂ is placed between the B and D ends. Clearly, the potential of point A will be equal to the potential of the positive plate of the cell and the potential of point C will be equal to the potential of the negative plate of the cell.

It is clear from the figure that the resistances P and Q are in series when the key K₂ is open. Similarly, resistances R and S are in series, but P and Q together (arm ABC) and R and S together (arm ADC) are connected in parallel to each other. Since the side BD of the galvanometer is placed like a bridge over the sides ABC and ADC of the quadrilateral, this circuit is called a bridge.

When the bridge is in an equilibrium state, that is, there is no deflection in the galvanometer. That is, in the equilibrium state of the bridge, the ratio of the resistances of any two adjacent arms is equal to the ratio of the remaining two adjacent sides. 

Wheatstone Bridge Derivation

Suppose, on pressing the cell key K₁, a current I flows through the cell, which splits into two parts at the end A. One part I₁ flows through the resistance P in arm AB and the other part I₂, through the resistance R in arm AD. The current I₁ again comes to end B and gets divided into two parts. One part of it I_g flows through the galvanometer in arm BD and the remaining part (I₁– I_g) flows through resistance Q in arm BC. At the end D, the current I₂ from arm AD and the current Ig from arm BD, so the current flowing through the resistance S in arm DC will be (I₂ + I_g).

So according to Kirchhoff’s law, in closed path ABDA,

I₁P + I_gG – I₂R = 0            . . . (1)

And in closed path BCDB, 

(I₁ – I_g)Q – (I₂ + I_g)S – I_gG = 0          . . . (2)

The values ​​of the resistors P, Q, R, and S are taken in such a way that no current flows through the galvanometer G when the key K₂, is pressed. This is called the equilibrium state of the bridge, that is, in the equilibrium state of the bridge, the deflection in the galvanometer is zero (I_g = 0).

Putting I_g = 0, in the above equations,

I₁P = I₂R and I₁Q = I₂S

or 

I₁P / I₁Q = I₂R / I₂S

or 

P / Q = R / S

This is the necessary condition for the balance of the Wheatstone Bridge. With the help of the above formula, knowing the values ​​of three resistors P, Q, and R, the value of the fourth resistance S can be found.

Wheatstone Bridge Formula

The Wheatstone Bridge Formula for the calculation of the unknown resistor is as follows:

R = PS/Q

where, 
P and Q is the resistance of ratio arm
S is the known resistance of the standard arm
R is the unknown resistance

Advantages of Wheatstone’s Bridge

Various advantages of the Wheatstone’s Bridge are,

• With the help of Wheatstone’s Bridge, we can build a Meter bridge.
• The biggest advantage of Wheatstone’s Bridge is to accurately measure the electric resistance instead of using costly instruments.
• We can measure minute changes in the bridge, even in m ohm.
• It is very easy to find out the unknown resistance as the rest of the three are easily known.
• We can measure strain and pressure using a Wheatstone bridge.

Disadvantages of Wheatstone’s Bridge

Various disadvantages of the Wheatstone’s Bridge are,

• The result of Wheatstone’s Bridge can be easily affected by temperature and EMF cells.
• Wheatstone bridge may also get affected if the galvanometer is not of good quality.
• Wheatstone Bridge fails if it is not in a balanced condition.
• We can’t measure large resistance with the help of Wheatstone’s Bridge. 
• The cost of maintaining the Wheatstone Bridge is very high.

Wheatstone Bridge Applications

Wheatstone Bridge and the modification of Wheatstone Bridge are very useful tools in the field of physics and are used in a variety of measurement use cases, some of which are as follows:

• The most common use of Wheatstone Bridge is to measure resistance,  as it can measure resistance very precisely. A Meter Bridge is one of the applications of Wheatstone Bridge, which can measure unknown resistance with the help of everyday materials and a galvanometer.
• With some modifications in Wheatstone Bridge, we can even measure quantities like Capacitance, Inductance, and Impedance.
• Using Wheatstone Bridge and operational amplifiers, we can measure a variety of physical parameters such as temperature strain, light, etc.

Wheatstone Bridge Limitations

Wheatstone Bridge is one of the best tools but it also has major limitations, which are as follows:

• Wheatstone Bridge is not ideal for very low resistance measurement as the resistance of contact and leads also has some amount of resistance, which can introduce a significant amount of errors in the measurement. Kelvin’s Double Bridge (a modification of Wheatstone Bridge) is used for the measurement of small resistance.
• For High resistance measurements (mega ohms and giga ohms of resistance), the resistance provided by the bridge becomes so high that the galvanometer becomes sensitive to imbalance. So, we can’t use Wheatstone Bridge in case of high resistance measurement.
• The resistance of a conductor depends on the temperature and the heating effect of the current causes the conductor to heat and heat changes the resistance of the conductor. This can introduce errors in the calculation and for excessive amounts of current the error can be very significant. Thus, it is also a flaw in the design of Wheatstone Bridge.

Sample Questions on Wheatstone Bridge

Question 1: Find the equivalent resistance between points A and C in the circuit shown in the figure below:

Answer: 

The equivalent circuit of the circuit shown in the above figure is given as:

Since, 2/3 = 4/6

This is the circuit of a balanced Wheatstone bridge.  

In the balanced state, V_B = V_D  (where V represents potential) 

So no current will flow through the 5 Ω resistance.  

Now the equivalent resistance of sides AB and BC is R’ = 2 + 3 = 5 Ω.  

The equivalent resistance of AD and DC arm R” = 4 + 6 = 10 Ω 

If the equivalent resistance between the points A and C is R, and R is parallel combination of resistance R’ and R”,

⇒  1 / R = (1 / R’) + (1 / R”)

⇒  1/R = (1/5) + (1/10)

⇒  1/R = (2 + 1) / 10

⇒  R = 10/3 

⇒  R = 3.33 Ω

Question 2: The electric circuit of a balanced Wheatstone bridge is shown in Figure. Calculate the resistance x.

Answer:

Let the total resistance in arm BC be R. Since the bridge is balanced, therefore:

15/R = 5/10 

⇒ R = (15 × 10)/5 = 30 Ω 

Now, as R is parallel combination of 60 Ω and X.

⇒  1/R = (1/X) + (1/60)

⇒  1/30 = 1/X + (1/60)

⇒  1/X = 1/30 – 1/60

⇒  1/X = (2 – 1)/60 = 1 / 60

⇒  X = 60 Ω 

Question 3: What is a Meter Bridge and what kind of precautions do we need to perform measurements using a Meter Bridge?

Answer:

A Meter bridge is a device based on the principle of the Wheatstone bridge, with the help of which the resistance and specific resistance of a conductor can be determined. In this, a 1-meter long wire acts as the proportional side.

Precautions while using Meter Bridge are: 

• The ends of all the connection wires should be cleaned with sandpaper.  
• The current should not flow in the circuit for a long time otherwise, its resistance increases due to the heating of the bridge wire. Therefore, the key in the cell circuit should be plugged in only when observations are to be made. 
• The jockey should not be run by rubbing it on the meter bridge wire otherwise, the thickness of the wire will not remain the same at all places.  
• A shunt with a galvanometer should be used initially while adjusting, but the shunt should be removed near the position of zero deflection.  
• Only such a resistance plug should be removed from the resistance box so that the position of zero deflection is approximately in the middle of the bridge wire. In this case, the sensitivity of the bridge is maximum and the percentage error is minimum.  
• All other plugs in the resistance box, except those that have been removed, should be tightly packed. 

Question 4: In the following figure, find the current through the 4Ω resistor.

Answer:

Since, Q / P = S / R  

⇒ 4 / 20 =10 / 50 

It is an example of balanced wheat-stone bridge.

So, No current will flow through 16 Ω resistance.

As we know that current divide in inverse ratio, current through 4Ω resistance is,

= 1.4 × (20+4)+(50+10) / (50+10) 

= 1.4 × (7/5) 

= 1 A

FAQs on Wheatstone Bridge

Q1: What is Wheatstone Bridge?

Answer:

A circuit is designed for the measurement of an unknown resistance by connecting three known and one unknown resistance in the form of a quadrilateral, and a voltage is applied to the opposite corners of the quadrilateral. This is known as the Wheatstone Bridge.

Q2: What is the principle of Wheatstone Bridge?

Answer:

The Wheatstone Bridge works on the principle of null deflection, where the ratio of the resistance is equal and no current flows through the galvanometer.

Q3: What is the balanced condition of the Wheatstone Bridge?

Answer:

When no current flows through the galvanometer and it shows no deflection, the Wheatstone Bridge is said to be in a balanced condition. This condition is also called the equilibrium condition as well.

Q4: When is Wheatstone Bridge most sensitive?

Answer:

When all four resistances in the wheatstone bridge are almost equal, then the wheatstone bridge is said to be most sensitive.

Q5: Which instrument is used as a null detector in Wheatstone Bridge?

Answer:

The galvanometer is used as a null detector in the Wheatstone Bridge, where a null point is the condition where no current passes through the circuit where the galvanometer is connected.