Wheatstone bridge formula Class 12 derivation notes application balance condition principle

find Wheatstone bridge formula Class 12 derivation notes application balance condition principle ?

Electric Current

The rate of flow of charge is called electric current. If the rate of flow of charge does not change with time, the current is said to be steady or constant [Fig. below (a)]. For steady current,

I = q/t

where q is the amount of charge flowing through any cross-section of the conductor in time t.

In many situations, the current may vary with time [Fig. (b) and Fig. (c)]. In such situations, the current is given by

I = dq/dt

Convention Regarding Direction of Current

In metallic conductors, the charge is carried by free electrons. In electrolytes, the charge is carried by positive and negative ions. The direction of current is taken to be the direction in which the positive charge moves. A positive charge moving in one direction is equivalent to negative charge moving in the opposite direction (Fig. below).

Drift Speed of Electrons in a Conductor

A metallic conductor has a large number of free electrons. When a potential difference is applied across the ends of a conductor, an electric field is set up in the conductor which accelerates the electrons in the direction opposite to the direction of the electric field. Due to collisions with the atoms of the conductor, these electrons move with a velocity called the drift velocity  (v_d ) which is defined as the average velocity of the free electrons of the conductor under the influence of the electric field (E) and is given by

where Τ is the average time between two successive collisions and is called the relaxation time. The negative sign indicates that the electrons drift in a direction opposite to that of the field.

Relations Between Drift Speed and Current

The drift speed is related to current I in a conductor as

where n = number of free electrons per unit volume,
e = magnitude of charge of an electron
A = cross-sectional area of conductor

note : For moderate electric field, the drift-speed of electrons in a conductor is of the of order of a few mms^–1

Ohm’s Law

Ohm’s Law states that the current flowing through a conductor is directly proportional to the potential difference across its ends, provided the physical conditions of the conductor remain the same. Thus,

V ∝ I or V = R I

where R is the resistance of the given conductor.

Electrical Resistivity

The resistance of a wire is directly proportional to its length (l) and inversely proportional to its area of cross section (A), i.e.

where ρ is a constant of proportionality and its value depends on the material of the wire. The constant ρ is called the resistivity of the material of the wire. Unit of ρ: From Eq. (1) we have

Hence ρ is measured in ohm metre (or Ω m). Note that the value of ρ is independent of the length and the area of cross-section of the wire; it depends only on the material of the wire and temperature. Thus, resistivity is a characteristic of the material. The reciprocal of resistivity is called conductivity.

Resistors in Series and Parallel

1. Resistors in series

When resistors are joined in series, the total resistance R of the combination is equal to the sum of the individual resistances, i.e.

R = R1 + R2 + R3 + …

The current is the same in all resistors. The total potential difference across the combination is equal to the sum of the potential differences across the individual resistors.

Resistors in parallel

When resistors are connected in parallel, the effective resistance of the combination is given by

The potential difference is the same across each resistor. The total current is equal to the sum of the currents in the individual resistors.

Emf, Terminal Voltage and Internal Resistance of a Cell

The potential difference between the terminals of a cell when it is on an open circuit, i.e. when no current is drawn from it is called its emf (E). The potential difference between the terminals of a cell when it is in a closed circuit, i.e. when a current is drawn from it is called its terminal voltage (V). Every cell has a resistance of its own called its internal resistance (r). The value of r depends upon the nature of electrodes, the nature of the electrolyte, size of electrodes and the distance between them. Figure below shows a cell of emf E, internal resistance r connected to an external resistance R.

Total resistance of the circuit = R + r. The current in the circuit is

Potential difference across r is v = Ir

Potential difference across R is V = IR. V is called the terminal voltage and v is the potential drop across the internal resistance

Thus E = V + v = I(R + r)
Terminal voltage is V = E – Ir

Grouping of Cells

(a) Cells connected in series Consider two cells of emfs E1 and E2 and internal resistances r1 and r2 connected in series as shown in Fig. below

The equivalent emf and equivalent internal resistance of the combination is given by

(b) Cells connected in parallel If the cells are connected as shown in Fig. below, then

1. If the individual cells have the same emf E, and the number of cells connected in series is n, the emf of the combination is nE and the total internal resistance is nr, where r is the internal resistance of each cell.

2. When identical cells are connected in parallel the total emf = E, the emf of any one of the cells. Here the sum of the reciprocals of the individual internal resistances is equal to the reciprocal of the total internal resistance.

3. Cells should be connected in series if the external resistance R is greater than internal resistance r.

4. Cells should be connected in parallel if R < r.

5. Mixed grouping of cells is employed if R is comparable with r. The current in the circuit when n cells, each of emf E, are connected in series is given by.

The current in the circuit when n cells, each of emf E, are connected in parallel is given by

If the arrangement consists of n rows of cells in parallel each having m cells in series, then

Kirchhoff’s Laws

First Law or Junction Rule

The algebraic sum of the currents at a junction in a circuit is zero. The currents entering the junction are taken as positive and those leaving the junction are taken as negative. Figure 12.19 shows a part of a circuit having a junction A.

i.e. sum of currents entering a junction = sum of currents leaving that junction.

Second Law or Loop Rule

In any closed circuit (or loop), the algebraic sum of the potential differences across the sources of current and across the resistances in the circuit (or loop) is zero. The sources of current are the emfs of the cells and potential differences across the resistance are the voltage drops (IR).

Sign Convention for emfs and Voltage drops

Consider the circuit shown in Fig. below. The circuit has three closed loops abefa, bcdeb and acdfa. To use the loop rule, follow the following steps.

1. Draw a arrow on the top of each cell pointing from the positive to the negative terminal.

2. Choose a closed loop and move in a clockwise direction in that loop.

3. While crossing a cell, if you are moving in the direction of the arrow drawn on the cell, the emf of the cell is taken as positive but if you are moving opposite to the direction of the arrow, the emf is taken as negative. Do not worry about the direction of the current in the branch of the circuit containing that cell.

4. While crossing a resistor, if you are moving in the direction of the current through that resistor, the potential drop (IR) across the resistor is taken as positive but if you are moving opposite to the direction of the current, the potential drop across the resistor is taken as negative. In the circuit shown in Fig. above, there are two junctions at b and e. Applying the junction rule at either b or e we have

I₁ + I₂ = I₃

Applying the loop rule to loops abefa and bcdeb we have

If the values of E₁, E₂, R₁, R₂ and R₃ are known, the values of I₁, I₂ and I₃ can be obtained by the simultaneous solutions of Eqs. (1), (2) and (3).

note : 1.  Select as many loops as the number of unknowns. In the circuit shown in Fig. above. There are two unknowns I₁ and I₂. The third unknown I3 is determined by using the junction rule [Eq. (1)]. So we select two out of the three loops.

2.  If the directions of currents are not given, choose any direction (clockwise or anticlockwise) in a loop and calculate the values of currents. If any current turns out to be negative, it indicates that our choice of the direction of that current is incorrect. So reverse the direction of that current. The magnitude of the current remains the same.

Wheatstone’s Bridge

The network of four resistances P, Q, R and S shown in Fig. below is called Wheatstone’s Bridge. A battery is connected between A and C and a galvanometer is connected between B and D. If the values of P, Q, R and S are such that no current flows through the galvanometer, the bridge is said to be balanced. Then points B and D are at the same potential.

Condition for balanced Wheatstone’s Bridge

The currents in the branches of the bridge are shown in the figure. Applying the loop rule to loops ABDA and BCDB, we have

This is the condition for a balanced bridge. Figure below shows a simple form of a Wheatstone’s Bridge, called the metre bridge.

The resistance of the wire of length AD = l₁ serves as the resistor R and the resistance of the wire of length DC = l2 serves as the resistor S. If the wire AC is uniform, the resistances of the parts AD and DC of the wire will be proportional to their lengths l₁ and l₂. For a balanced metre bridge, we have

The Potentiometer

Figure below shows a potentiometer where AB is a wire of uniform cross-section, V is the driver battery and E is a cell. The emf of the battery is greater than that of the cell.

The principle of a potentiometer is based on the fact that the potential difference across any length of the wire is proportional to that length of the wire. If R is the resistance of the potentiometer wire AB and L is its length, then R = ρL/A where ρ is the resistivity of the material of the wire and A its cross-sectional area. From Ohm’s law, V = IR or V = IρL/A = KL where K = Iρ/A, which is constant for a given wire. Hence

V ∝ L

or V/L = K = potential gradient or fall of potential per unit length of the potentiometer wire. Hence potential difference v across any portion l of the wire is v = Kl. The galvanometer will show no deflection if v = emf E of the cell E or E = v = Kl. The length l is called the balancing length of the potentiometer wire. At balance point, potential difference across l due to the driver cell V = emf of cell E

Applications of Potentiometer

1. Comparison of emfs of two cells If l₁ and l₂ are the balancing lengths will cells of emfs E1 and E2, then

2. Determination of internal resistance of a cell If l₁ is the balancing length will cell of emf E when switch S is open [Fig. below], then

E = K l₁

A known resistance R is connected across the cell E, switch S is closed and the new balancing length l₂ is found, then (since a current is now drawn from cell E), the terminal voltage V of E is equal to Kl₂, i.e.

Now E = V + v = IR + Ir where I is the current drawn from E and r is its internal resistance. Thus

Knowing the values of R,l₁ and R₂, the value of r is determined.

Finding Equivalent Resistance by Using Symmetry in Networks

Let us consider a balanced Wheatstone’s bridge again as shown in Fig. below (a).

Notice that the network is symmetric about the line XY. By symmetry, we mean that the circuit to the right of XY is just the mirror image of the circuit to the left of XY. In such networks, the potentials and currents must also by symmetrical. Therefore, the current in branch ab is equal to the current in branch bc. Also the current in ad is equal to the current in dc. Furthermore, the potential at b is equal to the potential at d. Hence no current flows in the resistor between b and d (i.e., the resistor which lies in the line of symmetry). This is called the symmetry condition. Since no current flows through resistor R between b and d, this resistor is ineffective and can be removed. So the circuit simplifies to that shown in Fig. above (b). The equivalent resistance between points A and B is

The same symmetry condition applies also to a network of capacitors as shown in Fig. below.

The equivalent capacitance between points A and B is

Now let us consider a more complicated network as shown in Fig. below (a).

In this case, resistances between b and e and between e and d are ineffective (because these resistances lie along the line XY of symmetry) and the network simplifies to that shown in Fig. above (b). The equivalent resistance between A and B is

Fig. below shows the corresponding network of capacitors.

In this case, the equivalent capacitance between A and B is

Ammeter

Ammeter is used for measuring current in a circuit. It is a galvanometer having a very small resistance (called shunt) connected in parallel with it. The ammeter is always connected in series in the circuit the current through which is to be measured.

The range of an ammeter is the maximum current it can measure. The value of the shunt resistance determines the range of an ammeter. Figure above shows a part of a circuit, where

I = current to be measured,
G = galvanometer resistance,
Ig = current through the galvanometer for full scale
deflection, and
S = shunt resistance
It is clear that (since p.d. across G = p.d. across S)

Voltmeter

Voltmeter is used for measuring potential difference across a resistor in a circuit. It is a galvanometer having a very high resistance connected in series with it. The voltmeter is always connected in parallel with the resistor across which the potential difference is to be measured. The range of a voltmeter is the maximum voltage it can measure. The range is determined by the value of high resistance connected in series with it. Figure below shows a voltmeter, where

V = voltage to be measured,
Ig = current for full scale deflection
R = required high resistance.
It is clear that

Heating Effect of Current

If a current I amperes flows for time t seconds through a resistor R ohms across which a potential difference V volts is maintained, the amount of heat energy H (in joules) delivered is given by

To obtain heat in calories, H is divided by J = 4.2 joules per calorie is called the mechanical equivalent of heat. Electrical Power –

If E is the emf of a source of internal resistance r, the power delivered to an external circuit of resistance R (called the output power) is given by

Electrical Energy –

The SI unit of energy is joule. A practical unit for electrical energy is called kilowatt hour (kWh). Since watt = joule per second, watt second is the same as joule. Hence

1 watt hour = 1 watt x (60 x 60) seconds
= 3600 watt seconds
= 3600 joules
1 kilowatt hour = 1000 watt hours
= 1000 x 3600 joules
= 3.6 x 10⁶  joules

Note that watt and kilowatt are units of electrical power and watt hour and kilowatt hour are units of electrical energy.

 Power-Voltage Rating of Electrical Appliances

Every electrical appliance has a specified power-voltage (P – V) rating which determines the resistance of the appliance and the current it will draw. Since P = VI, the current that the appliance will draw is given by

I = P/V

The electrical wiring should be able to withstand this current. The resistance of the appliance is given by

1. Power of Electrical Appliances Connected in Parallel Let R1, R2, R3, … be the resistances of the electrical appliances meant to operate at the same voltage V and let P1, P2, P3, … be their respective electrical powers. Then

When the appliances are connected in parallel, their combined resistance R is given by

Total power consumed is

2. Power of Electrical Appliances Connected in Series The total resistance R when the appliances are connected in series is given by

Variation of Resistance and Resistivity with Temperature

The resistance of a conductor (and hence its resistivity) depends upon its temperature. For metallic conductors, the resistance increases with temperature. For small changes in temperature, we have

where R1 is the resistance at a temperature t1, R2 is the resistance at temperature t2 and a is the temperature coefficient of resistance. The variation of resistivity with temperature is given by

For metallic conductors, the temperature coefficient of resistance is positive. Some materials, such as carbon and semi-conductors have a negative temperature coefficient of resistance. The resistance of such materials decreases with increase in temperature.