Notes-Part-2-Class-12-Physics-Chapter-9-Current Electricity-MSBSHSE

A) To Compare emf. of Cells :

A battery of stable emf E is used to set up a potential gradient V/L along a potentiometer wire, where V ≡ potential difference across the length L of the wire. The positive terminals of the cells, whose emf’s (E₁ and E₂) are to be compared, are connected to the high potential terminal A. The negative terminals of the cells are connected to a galvanometer G through a two-way key. The other terminal of the galvanometer is connected to a pencil jockey. The emf E should be greater than both the emf’s E₁ and E₂.

Connecting point P to C, the cell with emf E₁ is brought into the circuit. The jockey is tapped along the wire to locate the null point D at a distance l₁ from A. Then,

E₁ = l₁(V/ L)

Now, without changing the potential gradient (i.e., without changing the rheostat setting) point Q (instead of P) is connected to C, bringing the cell with emf E₂ into the circuit. Let its null point D’ be at a distance l₂ from A, so that

E₂ = l₂(V/ L)

∴ E₁/E₂ = l₁/l₂

Hence, by measuring the corresponding null lengths l₁ and l₂, E₁/E₂ can be calculated. The experiment is repeated for different potential gradients using the rheostat.

Remember : This method is used when the two emf’s have comparable magnitudes. Then, the errors of measurement of their balancing lengths will also be of comparable magnitudes.

Method II (sum and difference method ) :

The emfs of cells can be compared also by another method called sum and difference method.

• When two cells are connected so that the positive terminal of the first cell is connected to the negative terminal of the second cell as shown in Fig (a). The emf of the two cells are added up and the effective emf of the combination of two cells is E₁ + E₂ . This method of connecting two cells is called the sum method. Here, the cells assist each other.
• When two cells are connected so that their negative terminals are together or their positive terminals are connected together as shown in Fig. (b). In this case their emf oppose each other and effective emf of the combination of two cells is E₁ – E₂ (E₁ > E₂ assumed). This method of connecting two cells is called the difference method. Remember that this combination of cells is not a parallel combination of cells.

Use of potentiometer to compare the emf's of two cells by the combination method (the sum and difference method) :

Explanation :

A battery of stable emf E is used to set up a potential gradient V/L, along the potentiometer wire, where V ≡ potential difference across length L of the wire.

The positive terminal of the cell 1 is connected to the higher potential terminal A of the potentiometer; the negative terminal is connected to the galvanometer G through the reversing key. The other terminal of the galvanometer is connected to a pencil jockey.

The cell 2 is connected across the remaining two opposite terminals of the reversing key. The other terminal of the galvanometer is connected to a pencil jockey.

The emf E₁ should be greater than the emf E₂; this can be adjusted by trial and error.

Two plugs are inserted in the reversing key in positions 1—1. Here, the two cells assist each other so that the net emf is E₁ + E₂. The jockey is tapped along the wire to locate the null point D. If the null point is a distance l₁ from A,

E₁ + E₂ = l₁ (V/L)

For the same potential gradient (without changing the rheostat setting), the plugs are now inserted into position 2—2. (instead of 1—1).

The emf E₂ then opposes E₁ and the net emf is E₁ − E₂.

The new null point D’ is, say, a distance l₂ from A and

E₁ − E₂ = l₂ (V/L)

∴ \(\frac{E_1+E_2}{E_1-E_2}=\frac{l_1}{l_2}\)

∴ \(\frac{E_1}{E_2}=\frac{l_1+l_2}{l_1-l_2}\)

Here, the emf E should be greater than E₁ + E₂

The experiment is repeated for different potential gradients using the rheostat.

Remember : This method is used when E₁ >> E₂, So that E₁ + E₂ and E₁ − E₂ have comparable magnitudes. Then the errors of measurement of l₁ and l₂ will also be of comparable magnitudes.

B) To Find Internal Resistance (r) of a Cell:

The internal resistance of a cell is the resistance offered by the electrolyte and electrodes in the cell.

The internal resistance of a cell depends on :

• Nature of the electrolyte : The greater the conductivity of the electrolyte, the lower is the internal resistance of the cell.
• Separation between the electrodes : The larger the separation between the electrodes of the cell, the higher is the internal resistance of the cell. This is because the ions have to cover a greater distance before reaching an electrode.
• Nature of the electrodes.
• The internal resistance is inversely proportional to the common area of the electrodes dipping in the electrolyte.

Experimental set up for this method :

Principle : A cell of emf E and internal resistance r, which is connected to an external resistance R, has its terminal potential difference V less than its emf E.

If I is the corresponding current,

\(\frac{E}{V}=\frac{I(R+r)}{IR}\)=\(\frac{R+r}{R}\)

= \(1+\frac{r}{R}\)  ( when R → ∞, v→ E)

∴ r = \(\frac{E-V}{V}R\)

Working : A battery of stable emf E’ is used to set up a potential gradient V/L, along the potentiometer wire, where V ≡ potential difference across the length L of the wire.

The negative terminal is connected through a centre-zero galvanometer G to a pencil jockey. A resistance box R with a plug key K in series is connected across the cell.

Firstly, key K is kept open; then, effectively, R = ∞. The jockey is tapped on the potentiometer wire to locate the null point D. Let the null length

AD = l₁, so that

E = (V_AB/L)l₁

With the same potential gradient, and a small resistance R in the resistance box, ‘key K is closed.

The new null length AD’ = l₂ for the terminal potential difference V is found :

V = (V_AB /L)l₂

∴ E/V = \(\frac{l_1}{l_2}\)   ∴\(\frac{E-V}{V}\)=\(\frac{l_1-l_2}{l_2}\)=\(\frac{l_1}{l_2}-1\)

Now  r = \(\frac{E-V}{V}R\)

∴ r = \(R(\frac{l_1}{l_2}-1)\)

R, l₁ and l₂ being known, r can be calculated. The experiment is repeated with different potential gradients using the rheostat or with different values of R.

C) Application of potentiometer:

The applications of potentiometer discussed above are used in laboratory. Some practical applications of potentiometer are given below.

(i) Voltage divider : The potentiometer can be used as a voltage divider to change the output voltage of a voltage supply.

As shown in the Fig. potential V is set up between points A and B of a potentiometer wire. One end of a device is connected to positive point A and the other end is connected to a slider that can move along wire AB. The voltage V divides in proportion of lengths l₁ and l₂ as shown in the figure.

(ii) Audio control : Sliding potentiometers are commonly used in modern low-power audio systems as audio control devices. Both sliding (faders) and rotary potentiometers (knobs) are regularly used for frequency attenuation, loudness control and for controlling different characteristics of audio signals.

(iii) Potentiometer as a sensor : If the slider of a potentiometer is connected to the moving part of a machine, it can work as a motion sensor. A small displacement of the moving part causes changes in potential which is further amplified using an amplifier circuit. The potential difference is calibrated in terms of the displacement of the moving part.

(iv) To measure the emf (for this, the emf of the standard cell and potential gradient must be known).

(v) To compare the emf’s of two cells.

(vi) To determine the internal resistance of a cell.

Construction of a galvanometer :

• A galvanometer consists of a coil of a large number of turns of fine insulated copper wire wound on a rectangular nonconducting, nonmagnetic frame.
• The coil is pivoted (or suspended) between cylindrically concave pole pieces of a horseshoe strong permanent magnet.
• The coil swings freely around a cylindrical soft iron core fitted between the pole pieces.
• The deflection of the coil can be read with a pointer attached to it. The position of the pointer on the scale provided depends on the current passing through the galvanometer (or the potential difference across it). A galvanometer can be used as an ammeter or a voltmeter with a suitable modification.
• The galvanometer coil has a moderate resistance (about 100 ohms) and the galvanometer itself has a small current carrying capacity (about 1 mA).

Formula to convert a moving coil galvanometer (MCG ) into an ammeter :

The shunt resistance is calculated as follows. In the arrangement shown in the figure,

Let I be the maximum current to be measured and I_g the current for which the galvanometer of resistance G shows a full-scale deflection. Then, the shunt resistance S should be such that the remaining current I — I_g = I_s is shunted through it.

In the parallel combination, the potential difference across the galvanometer

= the potential difference across the shunt

 I_g G = I_s S

= (I − I_g) S

∴ S = \((\frac{I_g}{I-I_g})G\)   ……. (1)

This is the required resistance of the shunt. The scale of the galvanometer is then calibrated so as to read the current in ampere or its submultiples (mA. μA) directly.

(i) If the current I is n times current I_g, then I = nI_g. Using this in the above expression we get

S = \((\frac{I_g}{nI_g-I_g})G\)=\(\frac{G}{n-1}\)

This is the required shunt to increase the range n times.

where n = I/I_s is the range-multiplying factor, i.e., the current range of the galvanometer can be increased by a factor n by connecting a shunt whose resistance is smaller than the galvanometer resistance by a factor n − 1.

∴ n =\(\frac{G}{S}+1\)= \(\frac{G+S}{S}\)

If R_A is the resistance of the ammeter,

R_A = \(\frac{GS}{G+S}\)=\(\frac{G}{n}\)

(ii) Also if I_s is the current through the shunt resistance, then the remaining current (I – I_s) will flow through galvanometer. Hence

G(I – I_s) = SI_s

∴ GI – GI_s = SI_s

∴ SI_s + GI_s = GI

∴ \(\frac{l_s}{l}=\frac{G}{G+S}\)

This equation gives the fraction of the total current through the shunt resistance.

Formula to Convert a Moving Coil Galvanometer into a Voltmeter.

A moving-coil galvanometer is converted into a voltmeter by increasing its effective resistance by connecting a high resistance R_S in series with the galvanometer, see Fig. The series resistance is also useful for changing the range of any given voltmeter.

Let G be the resistance of the galvanometer coil and I_g the current required for a full-scale deflection.

Let V be the maximum potential difference to be measured. The value of the series resistance R_S should be such that when the potential difference applied across the instrument is V, the current through the galvanometer is I_g

.In the series combination, the potential difference V gets divided across the galvanometer (resistance, G) and the resistance R_S :

V = I_gG + I_gR_S = I_g(G + R_S)

∴ R_S = \(\frac{V}{I_g}-G\)

This is the required value of the series resistance.

The scale of the galvanometer is then calibrated so as to read the potential difference in volt or its submultiples, e.g., mV, directly.

(i) The maximum potential difference V_g that can be dropped across the galvanometer is V_g = I_gG. Therefore, the above expression for the series resistance may be rewritten as

R_S = \(\frac{VG}{I_gG}-G\) = \(\frac{VG}{V_g}-G\) =G(n-1)

where n = V/V_g is the range-multiplying factor, i.e., the voltage range of the galvanometer can be increased by a factor of n by connecting a series resistance which is (n—1) times the galvanometer resistance.

∴ n = \(\frac{V}{V_g}=\frac{(R_S+G)I_g}{GI_g}\)=\(\frac{R_S+G}{G}\)

Since the resistance of the voltmeter is R_V = R_S + G,

∴ n = R_V/G

∴ R_V = Gn

Seebeck Effect

If two different metals are joined to form a closed circuit (loop) and these junctions are kept at different temperatures, a small emf is produced and a current flows through the metals. This emf is called thermo emf this effect is called the Seebeck effect and the pair of dissimilar metals forming the junction is called a thermocouple. An antimony-bismuth thermo-couple is shown in a diagram.

For this thermo couple the current flows from antimony to bismuth at the cold junction. (ABC rule). For a copper-iron couple (see diagram) the current flows from copper to iron at the hot junction.

This effect is reversible. The direction of the current will be reversed if the hot and cold junctions are interchanged.

The thermo emf developed in a thermocouple when the cold junction is at 0⁰ C and the hot junction is at T°C is given by

E = αT + ½  βT²

Here α and  β are called the thermoelectric constants. This equation tells that a graph showing the variation of E with temperature is a parabola.