##### Class 12^{th} Physics Part I CBSE Solution

**Exercises**- The storage battery of a car has an emf of 12 V. If the internal resistance of the battery…
- A battery of emf 10 V and internal resistance 3 W is connected to a resistor. If the…
- Three resistors 1Ω, 2Ω and 3Ω are combined in series. What is the total resistance of the…
- If the combination is connected to a battery of emf 12 V and negligible internal…
- Three resistors 2 W, 4 W and 5 W are combined in parallel. What is the total resistance of…
- If the combination is connected to a battery of emf 20 V and negligible internal…
- At room temperature (27.0 °C) the resistance of a heating element is 100Ω. What is the…
- A negligibly small current is passed through a wire of length 15 m and uniform…
- A silver wire has a resistance of 2.1 W at 27.5 °C, and a resistance of 2.7 W at 100 °C.…
- A heating element using nichrome connected to a 230 V supply draws an initial current of…
- Determine the current in each branch of the network shown in Fig. 3.30:…
- In a metre bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end A,…
- Determine the balance point of the bridge above if X and Y are interchanged.…
- What happens if the galvanometer and cell are interchanged at the balance point of the…
- A storage battery of emf 8.0 V and internal resistance 0.5Ω is being charged by a 120 V dc…
- In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm…
- The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5…

**Additional Exercises**- The earth’s surface has a negative surface charge density of 10-9 C m-2. The potential…
- Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015Ω are…
- A secondary cell after long use has an emf of 1.9 V and a large internal resistance of…
- Two wires of equal length, one of aluminium and the other of copper have the same…
- What conclusion can you draw from the following observations on a resistor made of alloy…
- Answer the following questions: (a) A steady current flows in a metallic conductor of…
- Choose the correct alternative: (a) Alloys of metals usually have (greater/less)…
- Given n resistors each of resistance R, how will you combine them to get the (i) maximum…
- Given the resistances of 1Ω, 2Ω, 3Ω, how will be combine them to get an equivalent…
- Determine the equivalent resistance of networks shown in Fig. 3.31.…
- Determine the current drawn from a 12V supply with internal resistance 0.5Ω by the…
- Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω…
- Figure 3.34 shows a 2.0 V potentiometer used for the determination of internal resistance…

**Exercises**

- The storage battery of a car has an emf of 12 V. If the internal resistance of the battery…
- A battery of emf 10 V and internal resistance 3 W is connected to a resistor. If the…
- Three resistors 1Ω, 2Ω and 3Ω are combined in series. What is the total resistance of the…
- If the combination is connected to a battery of emf 12 V and negligible internal…
- Three resistors 2 W, 4 W and 5 W are combined in parallel. What is the total resistance of…
- If the combination is connected to a battery of emf 20 V and negligible internal…
- At room temperature (27.0 °C) the resistance of a heating element is 100Ω. What is the…
- A negligibly small current is passed through a wire of length 15 m and uniform…
- A silver wire has a resistance of 2.1 W at 27.5 °C, and a resistance of 2.7 W at 100 °C.…
- A heating element using nichrome connected to a 230 V supply draws an initial current of…
- Determine the current in each branch of the network shown in Fig. 3.30:…
- In a metre bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end A,…
- Determine the balance point of the bridge above if X and Y are interchanged.…
- What happens if the galvanometer and cell are interchanged at the balance point of the…
- A storage battery of emf 8.0 V and internal resistance 0.5Ω is being charged by a 120 V dc…
- In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm…
- The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5…

**Additional Exercises**

- The earth’s surface has a negative surface charge density of 10-9 C m-2. The potential…
- Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015Ω are…
- A secondary cell after long use has an emf of 1.9 V and a large internal resistance of…
- Two wires of equal length, one of aluminium and the other of copper have the same…
- What conclusion can you draw from the following observations on a resistor made of alloy…
- Answer the following questions: (a) A steady current flows in a metallic conductor of…
- Choose the correct alternative: (a) Alloys of metals usually have (greater/less)…
- Given n resistors each of resistance R, how will you combine them to get the (i) maximum…
- Given the resistances of 1Ω, 2Ω, 3Ω, how will be combine them to get an equivalent…
- Determine the equivalent resistance of networks shown in Fig. 3.31.…
- Determine the current drawn from a 12V supply with internal resistance 0.5Ω by the…
- Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω…
- Figure 3.34 shows a 2.0 V potentiometer used for the determination of internal resistance…

###### Exercises

**Question 1.**The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4 W, what is the maximum current that can be drawn from the battery?

**Answer:**The diagram is given as:

Given: Electromotive force (E) of battery = 12V

Internal Resistance r of Battery = 0.4 Ω

Let the maximum current drawn from the battery be I.

Therefore, as per Ohm’s Law,

E = Ir

Hence maximum current drawn from battery is 30Ampere.

**Question 2.**A battery of emf 10 V and internal resistance 3 W is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed?

**Answer:**The diagram is given as:

Given: Electromotive force (E) of battery = 10 volt.

Internal Resistance r of the battery = 3 Ω

Current in the Circuit (I) = 0.5 A

Let the resistance of the required resistor is R Ω

Let the Terminal voltage of the register be V.

As per Ω ’s law,

⇒

⇒

⇒R + r = 20 Ω

∵ r = 3Ω

Resistance of Resistor is 17 Ω.

Also, as per Ohm’s law,

⇒V = IR

⟹ V = 0.5A × 17Ω =

⇒ V = 8.5V

∴ Terminal voltage of the register is 8.5V.

**Question 3.**Three resistors 1Ω, 2Ω and 3Ω are combined in series. What is the total resistance of the combination?

**Answer:**The diagram is given as:

When three resistors are connected in series then the total resistance is equal to sum of individual resistance.

Total resistance is equal to = 1 + 2 + 3 = 6 Ω

**Question 4.**If the combination is connected to a battery of emf 12 V and negligible internal resistance, obtain the potential drop across each resistor.

**Answer:**Let the current flowing in the circuit be I.

Electromotive force EMF, (E) = 12V

Total resistance in the circuit R = 6 Ω

According to Ohm ’s law, V = IR

Potential drop across 2 Ω resistor = 2 × 2 = 4Volt, as V = I × R by Ohm ’s Law

Potential drop across 1 Ω resistor = 2 × 1 = 2Volt

Potential drop across 3 Ω resistor = 2 × 3 = 6Volt

**Question 5.**Three resistors 2 W, 4 W and 5 W are combined in parallel. What is the total resistance of the combination?

**Answer:**

Since three resistors are connected in parallel, hence total resistance(R) of the combination is given by

Hence, Total Resistance is 1.05 Ω

**Question 6.**If the combination is connected to a battery of emf 20 V and negligible internal resistance, determine the current through each resistor, and the total current drawn from the battery.

**Answer:**Given, EMF of the battery E = 20Volt

Current flowing through 2 Ω resistor is given by

Current flowing through 4 Ω resistor is given by

Current flowing through 5 Ω resistor is given by

Current flowing through 2 Ω resistor is 10Ampere; 4 Ω resistor is 5Ampere and through 5 Ω resistor is 4Ampere respectively.

**Question 7.**At room temperature (27.0 °C) the resistance of a heating element is 100Ω. What is the temperature of the element if the resistance is found to be 117Ω, given that the temperature coefficient of the material of the resistor is 1.70 × 10^{–4} °C^{–1}.

**Answer:**Given: Temperature coefficient of filament,

α = 1.70 × 10^{-4} per °C

Let T_{1} be the temperature of element, R_{1} = 100Ω (Given: T_{1} = 27°C)

Let T_{2} be the temperature of element, R_{2} = 117Ω

To find T_{2} = ?

The formula is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ T_{2}-27°C = 1000°C

⇒ T_{2} = 1000°C + 27°C

Hence Temperature of element, T_{2} is 1027 °C.

__Note: The temperature coefficient of resistance (α) gives the change in resistance per degree change in temperature. For pure metals, it is positive means- The resistance increases with temperature.__

**Question 8.**A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 × 10^{–7} m^{2}, and its resistance is measured to be 5.0 W. What is the resistivity of the material at the temperature of the experiment?

**Answer:**Given, the length of wire l = 15 metre

Area of cross section of wire a = 6.0 × 10^{-7} metre square

Resistance of material of wire, R = 5 Ω

Resistivity of material of wire = ρ

Hence Resistivity of material of wire is 2 × 10^{-7} metre

**Question 9.**A silver wire has a resistance of 2.1 W at 27.5 °C, and a resistance of 2.7 W at 100 °C. Determine the temperature coefficient of resistivity of silver.

**Answer:**Resistance of silver wire at 27.5 °C R_{1} = 2.1 Ω

Resistance of silver wire at 100 ° C R_{2} = 2.7 Ω

Let temperature coefficient of silver wire is given by α.

The formula for α is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ α = 0.0039°C

Temperature Coefficient of silver wire is 0.0039 per °C.

**Question 10.**A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to due to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10^{–4} °C^{–1}.

**Answer:**Given Supply Voltage = 230 Volt

And it draws current I_{1} = 3.2 Ampere

Resistance is given by R_{1} = V/I

R_{1} = 230V/3.2A

R_{1} = 71.87 Ω

Steady value of current I_{2} = 2.8Ampere

Resistance R α is given by R_{2} = 230V/2.8A

R_{2} = 82.14 Ω

Temperature coefficient of nichrome wire α = 1.70 × 10^{-4} per °C

Initial temperature of nichrome T_{1} = 27 ° C

Steady Temperature attained by nichrome = T_{2}

The formula is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ T_{2}-27°C = 840.5°C

⇒ T_{2} = 840.5°C + 27°C = 867.5° C

Hence Steady Temperature attained by nichrome = 867.5 ° C.

**Question 11.**Determine the current in each branch of the network shown in Fig. 3.30:

**Answer:**The current through the labelled diagram shows the current flowing the respective branches:

For the closed circuit ABDA, potential is zero i.e.,

10I_{2} + 5I_{4} − 5I_{3} = 0

2I_{2} + I_{4} −I_{3} = 0

I_{3} = 2I_{2} + I_{4} ... (1)

For the closed circuit BCDB, potential is zero i.e.,

5(I_{2} − I_{4}) − 10(I_{3} + I_{4}) − 5I_{4} = 0

5I_{2} + 5I_{4} − 10I_{3} − 10I_{4} − 5I_{4} = 0

5I_{2} − 10I_{3} − 20I_{4} = 0

I_{2} = 2I_{3} + 4I_{4} ... (2)

For the closed circuit ABCFEA, potential is zero i.e.,

−10 + 10 (I1) + 10(I2) + 5(I2 − I4) = 0

10 = 15I_{2} + 10I_{1} − 5I_{4}

3I_{2} + 2I_{1} − I_{4} = 2 ... (3)

From equations (1) and (2), we obtain

I_{3} = 2(2I_{3} + 4I_{4}) + I_{4}

I_{3} = 4I_{3} + 8I_{4} + I_{4}

− 3I_{3} = 9I_{4}

− 3I_{4} = + I_{3} ... (4)

Putting equation (4) in equation (1), we obtain

I_{3} = 2I_{2} + I_{4}

− 4I_{4} = 2I_{2}

I_{2} = − 2I_{4} ... (5)

It is evident from the given figure that,

I_{1} = I_{3} + I_{2} ... (6)

Putting equation (6) in equation (3), we obtain

3I_{2} + 2(I_{3} + I_{2}) − I_{4} = 2

5I_{2} + 2I_{3} − I_{4} = 2 ... (7)

Putting equations (4) and (5) in equation (7), we obtain

5(−2 I_{4}) + 2(− 3 I_{4}) − I_{4} = 2

− 10I_{4} − 6I_{4} − I_{4} = 2

17I_{4} = − 2

I_{4} = -2/17Ampere

Equation (4) reduces to

I_{3} = − 3(I_{4})

I_{3} = -3 × -2/17

I_{3} = 6/17Ampere

I_{2} = -2(I_{4})

I_{2} = -2 × -2/17

I_{2} = 4/17Ampere

I_{2} - I_{4} = 6/17Ampere

I_{3} + I_{4} = 6/17Ampere

I_{1} = I_{2} + I_{3} = 4/17 + 6/17 = 10/17 Ampere

Therefore, current in branch AB = 4/17Ampere

In branch BC = 6/17Ampere

In branch CD = -4/17Ampere

In branch AD = 6/17Ampere

In branch BD = -2/17Ampere

Total current = 4/17 + 6/17 + -4/17 + 6/17 + -2/17 = 10/17Ampere.

**Question 12.**In a metre bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end A, when the resistor Y is of 12.5Ω.

Determine the resistance of X. Why are the connections between resistors in a Wheatstone or meter bridge made of thick copper strips?

**Answer:**The diagram is shown below:

Balance Point from end A, l_{1} = 39.5 centimeter

Resistance of resistor Y = 12.5 Ω

Balance condition is given by,

⇒ X = 8.2 ohm

Thick copper strips are used to minimize the resistance of the connnecting wires

**Question 13.**Determine the balance point of the bridge above if X and Y are interchanged.

**Answer:**When X and Y gets interchanged then l1 and 100-l1 gets interchanged.

Now the balance point of bridge will be 100-l1 from A

100-l1 = 100-39.5 = 60.5cm

Hence the balance point is 60.5cm from A.

**Question 14.**What happens if the galvanometer and cell are interchanged at the balance point of the bridge? Would the galvanometer show any current?

**Answer:**When the galvanometer and cell are interchanged at balance point of bridge then no current would flow through galvanometer because of no deflection.

**Question 15.**A storage battery of emf 8.0 V and internal resistance 0.5Ω is being charged by a 120 V dc supply using a series resistor of 15.5Ω. What is the terminal voltage of the battery during charging? What is the purpose of having a series resistor in the charging circuit?

**Answer:**

Electromotive force EMF of battery = 8Volt

Internal resistance of battery r = 0.5 Ω

Supply Voltage V = 120Volt

Resistance of resistor R = 15.5 Ω

Effective voltage in circuit = V1

Since Resistance R is connected in series Hence We can write V1 = V-E

V_{1} = 120V-8V = 112 V

Current flowing in the circuit:

⇒

⇒

⇒

I = 7Ampere

By Ohm ’s Law, Voltage across resistor R is given by V = IR

V = 7A × 15.5Ω

⇒V = 108.5Volt

Supply Voltage = Terminal Voltage of battery + Voltage drop across Resistor R

Therefore Terminal Voltage of battery = 120 V-108.5 V = 11.5Volt

Series resistor in charging circuit reduces the current drawn from external supply and current will be too high in its absence.

**Question 16.**In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63.0 cm, what is the emf of the second cell?

**Answer:**Given EMF of the cell E1 = 1.25Volt

Balance point of the potentiometer l1 is given 35cm

Since the cell is replaced by another cell of EMF E2

New balance point of potentiometer l2 is 63cm

Since balance condition is given by

E2 = 1.25V × (63m/35m)

E2 = 2.25Volt.

**Question 17.**The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 × 10^{28} m^{–3}. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 × 10^{–6} m^{2} and it is carrying a current of 3.0 A.

**Answer:**Number density of free electrons in a copper conductor, n = 8.5 × 10^{28} per meter cube

Length of Copper Wire l = 3 meter

Area of cross section of wire A = 2 × 10^{-6} meter square

Current drawn by wire is given 3 Ampere

Since I = nAeV_{d}

Here V_{d} is the drift velocity and “e” is the electron charge 1.6 × 10^{-19} Coulomb

And V_{d} Drift velocity is

Therefore

I = nAel/t

Or t = nAel/I

⇒ t = 2.7 × 10^{4} second

**Question 1.**

The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4 W, what is the maximum current that can be drawn from the battery?

**Answer:**

The diagram is given as:

Given: Electromotive force (E) of battery = 12V

Internal Resistance r of Battery = 0.4 Ω

Let the maximum current drawn from the battery be I.

Therefore, as per Ohm’s Law,

E = Ir

Hence maximum current drawn from battery is 30Ampere.

**Question 2.**

A battery of emf 10 V and internal resistance 3 W is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed?

**Answer:**

The diagram is given as:

Given: Electromotive force (E) of battery = 10 volt.

Internal Resistance r of the battery = 3 Ω

Current in the Circuit (I) = 0.5 A

Let the resistance of the required resistor is R Ω

Let the Terminal voltage of the register be V.

As per Ω ’s law,

⇒

⇒

⇒R + r = 20 Ω

∵ r = 3Ω

Resistance of Resistor is 17 Ω.

Also, as per Ohm’s law,

⇒V = IR

⟹ V = 0.5A × 17Ω =

⇒ V = 8.5V

∴ Terminal voltage of the register is 8.5V.

**Question 3.**

Three resistors 1Ω, 2Ω and 3Ω are combined in series. What is the total resistance of the combination?

**Answer:**

The diagram is given as:

When three resistors are connected in series then the total resistance is equal to sum of individual resistance.

Total resistance is equal to = 1 + 2 + 3 = 6 Ω

**Question 4.**

If the combination is connected to a battery of emf 12 V and negligible internal resistance, obtain the potential drop across each resistor.

**Answer:**

Let the current flowing in the circuit be I.

Electromotive force EMF, (E) = 12V

Total resistance in the circuit R = 6 Ω

According to Ohm ’s law, V = IR

Potential drop across 2 Ω resistor = 2 × 2 = 4Volt, as V = I × R by Ohm ’s Law

Potential drop across 1 Ω resistor = 2 × 1 = 2Volt

Potential drop across 3 Ω resistor = 2 × 3 = 6Volt

**Question 5.**

Three resistors 2 W, 4 W and 5 W are combined in parallel. What is the total resistance of the combination?

**Answer:**

Since three resistors are connected in parallel, hence total resistance(R) of the combination is given by

Hence, Total Resistance is 1.05 Ω

**Question 6.**

If the combination is connected to a battery of emf 20 V and negligible internal resistance, determine the current through each resistor, and the total current drawn from the battery.

**Answer:**

Given, EMF of the battery E = 20Volt

Current flowing through 2 Ω resistor is given by

Current flowing through 4 Ω resistor is given by

Current flowing through 5 Ω resistor is given by

Current flowing through 2 Ω resistor is 10Ampere; 4 Ω resistor is 5Ampere and through 5 Ω resistor is 4Ampere respectively.

**Question 7.**

At room temperature (27.0 °C) the resistance of a heating element is 100Ω. What is the temperature of the element if the resistance is found to be 117Ω, given that the temperature coefficient of the material of the resistor is 1.70 × 10^{–4} °C^{–1}.

**Answer:**

Given: Temperature coefficient of filament,

α = 1.70 × 10^{-4} per °C

Let T_{1} be the temperature of element, R_{1} = 100Ω (Given: T_{1} = 27°C)

Let T_{2} be the temperature of element, R_{2} = 117Ω

To find T_{2} = ?

The formula is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ T_{2}-27°C = 1000°C

⇒ T_{2} = 1000°C + 27°C

Hence Temperature of element, T_{2} is 1027 °C.

__Note: The temperature coefficient of resistance (α) gives the change in resistance per degree change in temperature. For pure metals, it is positive means- The resistance increases with temperature.__

**Question 8.**

A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 × 10^{–7} m^{2}, and its resistance is measured to be 5.0 W. What is the resistivity of the material at the temperature of the experiment?

**Answer:**

Given, the length of wire l = 15 metre

Area of cross section of wire a = 6.0 × 10^{-7} metre square

Resistance of material of wire, R = 5 Ω

Resistivity of material of wire = ρ

Hence Resistivity of material of wire is 2 × 10^{-7} metre

**Question 9.**

A silver wire has a resistance of 2.1 W at 27.5 °C, and a resistance of 2.7 W at 100 °C. Determine the temperature coefficient of resistivity of silver.

**Answer:**

Resistance of silver wire at 27.5 °C R_{1} = 2.1 Ω

Resistance of silver wire at 100 ° C R_{2} = 2.7 Ω

Let temperature coefficient of silver wire is given by α.

The formula for α is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ α = 0.0039°C

Temperature Coefficient of silver wire is 0.0039 per °C.

**Question 10.**

A heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few seconds to due to a steady value of 2.8 A. What is the steady temperature of the heating element if the room temperature is 27.0 °C? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10^{–4} °C^{–1}.

**Answer:**

Given Supply Voltage = 230 Volt

And it draws current I_{1} = 3.2 Ampere

Resistance is given by R_{1} = V/I

R_{1} = 230V/3.2A

R_{1} = 71.87 Ω

Steady value of current I_{2} = 2.8Ampere

Resistance R α is given by R_{2} = 230V/2.8A

R_{2} = 82.14 Ω

Temperature coefficient of nichrome wire α = 1.70 × 10^{-4} per °C

Initial temperature of nichrome T_{1} = 27 ° C

Steady Temperature attained by nichrome = T_{2}

The formula is: R_{2} = R_{1}[1 + α(T_{2}-T_{1})]

⇒ R_{2} - R_{1} = R_{1}α(T_{2} - T_{1})

⇒ R_{2} - R_{1} = R_{1}α (T_{2} - T_{1})

⇒

⇒ T_{2}-27°C = 840.5°C

⇒ T_{2} = 840.5°C + 27°C = 867.5° C

Hence Steady Temperature attained by nichrome = 867.5 ° C.

**Question 11.**

Determine the current in each branch of the network shown in Fig. 3.30:

**Answer:**

The current through the labelled diagram shows the current flowing the respective branches:

For the closed circuit ABDA, potential is zero i.e.,

10I_{2} + 5I_{4} − 5I_{3} = 0

2I_{2} + I_{4} −I_{3} = 0

I_{3} = 2I_{2} + I_{4} ... (1)

For the closed circuit BCDB, potential is zero i.e.,

5(I_{2} − I_{4}) − 10(I_{3} + I_{4}) − 5I_{4} = 0

5I_{2} + 5I_{4} − 10I_{3} − 10I_{4} − 5I_{4} = 0

5I_{2} − 10I_{3} − 20I_{4} = 0

I_{2} = 2I_{3} + 4I_{4} ... (2)

For the closed circuit ABCFEA, potential is zero i.e.,

−10 + 10 (I1) + 10(I2) + 5(I2 − I4) = 0

10 = 15I_{2} + 10I_{1} − 5I_{4}

3I_{2} + 2I_{1} − I_{4} = 2 ... (3)

From equations (1) and (2), we obtain

I_{3} = 2(2I_{3} + 4I_{4}) + I_{4}

I_{3} = 4I_{3} + 8I_{4} + I_{4}

− 3I_{3} = 9I_{4}

− 3I_{4} = + I_{3} ... (4)

Putting equation (4) in equation (1), we obtain

I_{3} = 2I_{2} + I_{4}

− 4I_{4} = 2I_{2}

I_{2} = − 2I_{4} ... (5)

It is evident from the given figure that,

I_{1} = I_{3} + I_{2} ... (6)

Putting equation (6) in equation (3), we obtain

3I_{2} + 2(I_{3} + I_{2}) − I_{4} = 2

5I_{2} + 2I_{3} − I_{4} = 2 ... (7)

Putting equations (4) and (5) in equation (7), we obtain

5(−2 I_{4}) + 2(− 3 I_{4}) − I_{4} = 2

− 10I_{4} − 6I_{4} − I_{4} = 2

17I_{4} = − 2

I_{4} = -2/17Ampere

Equation (4) reduces to

I_{3} = − 3(I_{4})

I_{3} = -3 × -2/17

I_{3} = 6/17Ampere

I_{2} = -2(I_{4})

I_{2} = -2 × -2/17

I_{2} = 4/17Ampere

I_{2} - I_{4} = 6/17Ampere

I_{3} + I_{4} = 6/17Ampere

I_{1} = I_{2} + I_{3} = 4/17 + 6/17 = 10/17 Ampere

Therefore, current in branch AB = 4/17Ampere

In branch BC = 6/17Ampere

In branch CD = -4/17Ampere

In branch AD = 6/17Ampere

In branch BD = -2/17Ampere

Total current = 4/17 + 6/17 + -4/17 + 6/17 + -2/17 = 10/17Ampere.

**Question 12.**

In a metre bridge [Fig. 3.27], the balance point is found to be at 39.5 cm from the end A, when the resistor Y is of 12.5Ω.

Determine the resistance of X. Why are the connections between resistors in a Wheatstone or meter bridge made of thick copper strips?

**Answer:**

The diagram is shown below:

Balance Point from end A, l_{1} = 39.5 centimeter

Resistance of resistor Y = 12.5 Ω

Balance condition is given by,

⇒ X = 8.2 ohm

Thick copper strips are used to minimize the resistance of the connnecting wires**Question 13.**

Determine the balance point of the bridge above if X and Y are interchanged.

**Answer:**

When X and Y gets interchanged then l1 and 100-l1 gets interchanged.

Now the balance point of bridge will be 100-l1 from A

100-l1 = 100-39.5 = 60.5cm

Hence the balance point is 60.5cm from A.

**Question 14.**

What happens if the galvanometer and cell are interchanged at the balance point of the bridge? Would the galvanometer show any current?

**Answer:**

When the galvanometer and cell are interchanged at balance point of bridge then no current would flow through galvanometer because of no deflection.

**Question 15.**

A storage battery of emf 8.0 V and internal resistance 0.5Ω is being charged by a 120 V dc supply using a series resistor of 15.5Ω. What is the terminal voltage of the battery during charging? What is the purpose of having a series resistor in the charging circuit?

**Answer:**

Electromotive force EMF of battery = 8Volt

Internal resistance of battery r = 0.5 Ω

Supply Voltage V = 120Volt

Resistance of resistor R = 15.5 Ω

Effective voltage in circuit = V1

Since Resistance R is connected in series Hence We can write V1 = V-E

V_{1} = 120V-8V = 112 V

Current flowing in the circuit:

⇒

⇒

⇒

I = 7Ampere

By Ohm ’s Law, Voltage across resistor R is given by V = IR

V = 7A × 15.5Ω

⇒V = 108.5Volt

Supply Voltage = Terminal Voltage of battery + Voltage drop across Resistor R

Therefore Terminal Voltage of battery = 120 V-108.5 V = 11.5Volt

Series resistor in charging circuit reduces the current drawn from external supply and current will be too high in its absence.

**Question 16.**

In a potentiometer arrangement, a cell of emf 1.25 V gives a balance point at 35.0 cm length of the wire. If the cell is replaced by another cell and the balance point shifts to 63.0 cm, what is the emf of the second cell?

**Answer:**

Given EMF of the cell E1 = 1.25Volt

Balance point of the potentiometer l1 is given 35cm

Since the cell is replaced by another cell of EMF E2

New balance point of potentiometer l2 is 63cm

Since balance condition is given by

E2 = 1.25V × (63m/35m)

E2 = 2.25Volt.

**Question 17.**

The number density of free electrons in a copper conductor estimated in Example 3.1 is 8.5 × 10^{28} m^{–3}. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 × 10^{–6} m^{2} and it is carrying a current of 3.0 A.

**Answer:**

Number density of free electrons in a copper conductor, n = 8.5 × 10^{28} per meter cube

Length of Copper Wire l = 3 meter

Area of cross section of wire A = 2 × 10^{-6} meter square

Current drawn by wire is given 3 Ampere

Since I = nAeV_{d}

Here V_{d} is the drift velocity and “e” is the electron charge 1.6 × 10^{-19} Coulomb

And V_{d} Drift velocity is

Therefore

I = nAel/t

Or t = nAel/I

⇒ t = 2.7 × 10^{4} second

###### Additional Exercises

**Question 1.**The earth’s surface has a negative surface charge density of 10^{-9} C m^{–2}. The potential difference of 400 kV between the top of the atmosphere and the surface results (due to the low conductivity of the lower atmosphere) in a current of only 1800 A over the entire globe. If there were no mechanism of sustaining atmospheric electric field, how much time (roughly) would be required to neutralize the earth’s surface? (This never happens in practice because there is a mechanism to replenish electric charges, namely the continual thunderstorms and lightning in different parts of the globe). (Radius of earth = 6.37 × 10^{6} m.)

**Answer:**Surface Charge density of earth d = 10^{-9} Coulomb per meter square

Current over the entire globe = 1800 Ampere

Radius of earth r = 6.37 × 10^{6} meter

Surface area of earth A = 4 × π × radius × radius

A = 4 × π × 6.37 × 10^{6} × 6.37 × 10^{6}

A = 5.09 × 10^{14} m^{2}

Charge on the earth surface q = d × A

q = 10^{-9} × 5.09 × 10^{14} m^{2}

q = 5.09 × 10^{5} Coulomb

Let the time taken to neutralize earth surface = t

Current

t = 5.09 × 10^{5}C/1800 A

t = 282.78 seconds.

**Question 2.**Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015Ω are joined in series to provide a supply to a resistance of 8.5Ω. What are the current drawn from the supply and its terminal voltage?

**Answer:**Number of secondary cells n = 6

Emf of each secondary cell E = 2Volt

Internal resistance of each cell r = 0.015 Ω

Resistance of resistor R = 8.5 Ω

Let current drawn from supply = I

⇒ I = 1.39Ampere

Hence current drawn from supply is 1.39Ampere

**Question 3.**A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?

**Answer:**Emf of secondary cell E = 1.9Volt

Internal resistance of cell r = 380 Ω

⇒ I = 1.9V/380Ω

⇒ I = 0.005Ampere

Hence maximum current drawn from cell is 0.005Ampere and since a large current is required to start the motor of a car and hence cell cannot be used to start the motor.

**Question 4.**Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. (ρ_{Al} = 2.63 × 10^{–8} Ωm, ρ_{Cu} = 1.72 × 10^{–8} Ωm, Relative density of Al = 2.7, of Cu = 8.9.)

**Answer:**Resistivity of aluminium, p_{1} = 2.63 × 10^{−8} Ω m

Relative density of aluminium, d_{1} = 2.7

Let l1 be the length of aluminium wire and m1 be its mass.

Resistance of the aluminium wire = R_{1}

Area of cross-section of the aluminium wire = A_{1}

Resistivity of copper, p_{2} = 1.72 × 10^{−8} Ω metre

Relative density of copper, d_{2} = 8.9

Let l2 be the length of copper wire and m2 be its mass.

Resistance of the copper wire = R2

Area of cross-section of the copper wire = A2

We can write,

R_{1} = (p_{1} × l_{1})/A_{1}

R_{2} = (p_{2} × l_{2})/A_{2}

Since R_{1} = R_{2}

(p_{1} × l_{1})/A_{1} = (p_{2} × l_{2})/A_{2}

And l_{1} = l_{2}

Therefore p_{1}/A_{1} = p_{2}/A_{2}

A_{1}/A_{2} = (2.63 × 10-8)/(1.72 × 10^{–8})

⇒ A_{1}/A_{2} = 2.63/1.72

Mass of the aluminium wire,

m_{1} = Volume × Density

= A_{1} × l_{1} × d_{1} = A_{1} l_{1} d_{1} .....1

Mass of the copper wire, m_{2} = Volume × Density

= A_{2} × l_{2} × d_{2} = A_{2} l_{2} d_{2} .......2

Dividing equation (1) by equation (2), we obtain

m_{1}/m_{2} = (A_{1} l_{1} d_{1})/(A_{2} l_{2} d_{2})

Since l_{1} = l_{2}

m_{1}/m_{2} = A_{1} d_{1}/A_{2} d_{2}

Since A_{1}/A_{2} = 2.63/1.72 calculated above

⇒ m_{1}/m_{2} = (2.63 × 2.7)/(1.72 × 8.9)

⇒ m1/m2 = 0.46

Since m1 is smaller than m2 Hence, aluminium is lighter than copper.

Since aluminium is lighter, it is preferred for overhead power cables over copper.

**Question 5.**What conclusion can you draw from the following observations on a resistor made of alloy manganin?

**Answer:**From the given information shown in the table, the ratio of voltage and current is a constant value and the value is equal to 19.7 Hence manganin is a Ω ic conductor.

According to Ohm’s law the ratio of voltage and current is resistance of conductor so Resistance of manganin is 19.7 Ω .

**Question 6.**Answer the following questions:

(a) A steady current flows in a metallic conductor of non-uniform cross-section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?

(b) Is Ohm’s law universally applicable for all conducting elements? If not, give examples of elements which do not obey Ω ’s law.

(c) A low voltage supply from which one needs high currents must have very low internal resistance. Why?

(d) A high tension (HT) supply of, say, 6 kV must have a very large internal resistance. Why?

**Answer:**(a) When a steady current flows in a metallic conductor of non-uniform cross section, the current flowing through the conductor is constant.

Since Current density, electric field and drift speed are inversely proportional to area of cross section therefore they are not constant.

(b) No, Ω ’s law is not applicable for all conducting elements. Example of non Ω ic conductor is Vacuum diode semiconductor.

(c) According to Ω ’s law V = I × R

If Voltage V is low, then Resistance R must be very low so that high current can be drawn from source.

(d) A high Tension supply must have a very large internal resistance so that it can limit the current from exceeding the safety limit.

**Question 7.**Choose the correct alternative:

(a) Alloys of metals usually have (greater/less) resistivity than that of their constituent metals.

(b) Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals.

(c) The resistivity of the alloy manganin is nearly independent of/ increases rapidly with increase of temperature.

(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of (10^{22}/10^{23}).

**Answer:**(a) Alloys of metals usually have greater resistivity than that of their constituent metals.

Alloys are the homogeneous mixture of two or more metals and due to which alloys have better conducting properties as compared to pure metals. Pure metals have large number of free electrons and simple atomic arrangement but in alloys these free electrons number as well as their atomic arrangement changes. Hence alloys have greater resistivity than metals.

(b) Alloys usually have lower temperature coefficient of resistance than pure metals.

The resistance change factor per °C of temp change is called the temperature coefficient. Coefficient approaching zero can be obtained by alloying certain metals. So we can say that alloys have lower temperature coefficient of resistance than pure metals.

(c) The resistivity of alloy, manganin, is nearly independent of increase of temperature.

The resistivity of alloys is high and atoms are not arranged in proper order and with the increase in temperature resistivity increases due to collision and random motion of atoms.

(d) The resistivity of a typical insulator is greater than that of a metal by a factor of order of 10^{22}.

In case of insulator, resistivity decreases with increase of temperature.

**Question 8.**Given n resistors each of resistance R, how will you combine them to get the (i) maximum (ii) minimum effective resistance?

What is the ratio of the maximum to minimum resistance?

**Answer:**Total number of resistors = n

Resistance of each resistor = R

(i)

When n resistors are connected in series, effective resistance R1is the maximum,

given by the product nR.

Hence, maximum resistance of the combination, R1 = nR

(ii)

When n resistors are connected in parallel, the effective resistance (R_{2}) is the minimum, given by the ratio: R/n

Hence, minimum resistance of the combination, R_{2} = R/n

(iii) The ratio of the maximum to the minimum resistance is,

⇒

**Question 9.**Given the resistances of 1Ω, 2Ω, 3Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω, (iii) 6 Ω, (iv) (6/11) Ω?

**Answer:**(i)

The resistance of the given resistors is,

R1 = 1 Ω , R2 = 2 Ω , R3 = 3 Ω

Connect 2 Ω and 1 Ω resistor in parallel and 3 Ω resistor in series to them

Equivalent resistance R’ is {(2 × 1)/(2 + 1)} + 3

Hence R’ is equal to 11/3 Ω

(ii)

Now connect 2 Ω and 3 Ω resistor in parallel and one Ω resistor in series to it

Equivalent Resistance R” = {(2 × 3)/(2 + 3)} + 1

R” = 11/5 Ω

(iii)

Now connect all three resistor 1 Ω 2 Ω and 3 Ω in series

Hence total resistance R”’ is 1 + 2 + 3 = 6 Ω

(iv)

Now connect all three resistors in parallel;

Equivalent Resistance is R”” = (1 × 2 × 3)/(1 × 2 + 2 × 3 + 3 × 1)

R”” = 6/11 Ω

**Question 10.**Determine the equivalent resistance of networks shown in Fig. 3.31.

**Answer:**(a)

It can be observed from the given circuit that in the first small loop, two resistors

of resistance 1 Ω each are connected in series.

Hence, their equivalent resistance = (1 + 1) = 2 Ω

It can also be observed that two resistors of resistance 2 Ω each are connected in series.

Hence, their equivalent resistance = (2 + 2) = 4 Ω.

Now the given circuit diagram can be explained as

Now draw 4 loops each having 2 resistors of 2 Ω and 4 Ω connected in parallel.

Hence Equivalent Resistance R_{e} of each loop is

⇒ Re = 4/3 Ω

Since all 4 loops will be connected in series and each loop will have 4/3 Ω resistance

Hence, equivalent resistance of the given circuit is 4 × 4/3 = 16/3 Ω .

(b)

From the given circuit that five resistors of resistance R each are connected in series.

Hence, equivalent resistance of the circuit = R + R + R + R + R = 5R

**Question 11.**Determine the current drawn from a 12V supply with internal resistance 0.5Ω by the infinite network shown in Fig. 3.32. Each resistor has 1Ω resistance.

**Answer:**The resistance of each resistor connected in the given circuit, R = 1 Ω

Equivalent resistance of the given circuit = R’

The network is infinite. Hence, equivalent resistance is given by the relation,

R’ = 2 + (R’/R’ + 1)

or R’^{2} -2R’ -2 = 0

On solving R’ = (2+√4+8)/2

(Negative value of R’ cannot be accepted)

R’ = 1 + √3 Ω

R’ = 1 + 1.73 = 2.73 Ω

Internal resistance of the circuit, r = 0.5 Ω

Hence, total resistance of the given circuit = 2.73 + 0.5 = 3.23 Ω

Supply voltage, V = 12 V

According to Ω ’s Law, current drawn from the source is given by the ratio, = 12/3.23 = 3.72 A

**Question 12.**Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf e and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(a) What is the value e?

(b) What purpose does the high resistance of 600 kW have?

(c) Is the balance point affected by this high resistance?

(d) Would the method work in the above situation if the driver cell of the potentiometer had an emf of 1.0V instead of 2.0V?

(e) Would the circuit work well for determining an extremely small emf, say of the order of a few mV (such as the typical emf of a thermo-couple)? If not, how will you modify the circuit?

**Answer:**(a) Constant emf of the given standard cell, E_{1} = 1.02 V

Balance point on the wire, l_{1} = 67.3 cm

A cell of unknown emf, ε, replaced the standard cell. Therefore, new balance point on the wire,

l = 82.3 cm

The relation connecting emf and balance point is,

E_{1}/l_{1} = E/l

E = l × E_{1}/l_{1}

E = 82.3 cm × 1.02 V/67.3 cm

E = 1.247 Volt.

The value of unknown emf is 1.247 V.

(b) The purpose of using the high resistance of 600 kΩ is to reduce the current through the galvanometer when the movable contact is far from the balance point.

(c) The balance point is not affected by the presence of high resistance.

(d) The point is not affected by the internal resistance of the driver cell.

(e) The method would not work if the driver cell of the potentiometer has an emf of 1.0 V instead of 2.0 V. This is because if the emf of the driver cell of the potentiometer is less than the emf of the other cell, then we cannot obtain balance point on the wire.

(f) The circuit would not work well for determining an extremely small emf. As the circuit would be unstable, the balance point would be close to end A. Hence, there would be a high percentage of error. The given circuit can be modified if a series resistance is connected with the wire AB. The potential drop across AB is slightly greater than the emf measured. The percentage error would be small.

**Question 13.**Figure 3.34 shows a 2.0 V potentiometer used for the determination of internal resistance of a 1.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5Ω is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. Determine the internal resistance of the cell.

**Answer:**Balance point for open circuit, l_{1} = 78.3 cm

Resistance of the external resistor, R = 9.5 Ω

Balance point for this resistor, l_{2} = 64.8 cm

The relation connecting internal resistance r and balance point is,

r=(78.3-64.8)/64.8× 9.5

=0.177 × 9.5

=1.685 Ω

Therefore, the value of the unknown resistance, X, is 1.685 Ω.

**Question 1.**

The earth’s surface has a negative surface charge density of 10^{-9} C m^{–2}. The potential difference of 400 kV between the top of the atmosphere and the surface results (due to the low conductivity of the lower atmosphere) in a current of only 1800 A over the entire globe. If there were no mechanism of sustaining atmospheric electric field, how much time (roughly) would be required to neutralize the earth’s surface? (This never happens in practice because there is a mechanism to replenish electric charges, namely the continual thunderstorms and lightning in different parts of the globe). (Radius of earth = 6.37 × 10^{6} m.)

**Answer:**

Surface Charge density of earth d = 10^{-9} Coulomb per meter square

Current over the entire globe = 1800 Ampere

Radius of earth r = 6.37 × 10^{6} meter

Surface area of earth A = 4 × π × radius × radius

A = 4 × π × 6.37 × 10^{6} × 6.37 × 10^{6}

A = 5.09 × 10^{14} m^{2}

Charge on the earth surface q = d × A

q = 10^{-9} × 5.09 × 10^{14} m^{2}

q = 5.09 × 10^{5} Coulomb

Let the time taken to neutralize earth surface = t

Current

t = 5.09 × 10^{5}C/1800 A

t = 282.78 seconds.

**Question 2.**

Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015Ω are joined in series to provide a supply to a resistance of 8.5Ω. What are the current drawn from the supply and its terminal voltage?

**Answer:**

Number of secondary cells n = 6

Emf of each secondary cell E = 2Volt

Internal resistance of each cell r = 0.015 Ω

Resistance of resistor R = 8.5 Ω

Let current drawn from supply = I

⇒ I = 1.39Ampere

Hence current drawn from supply is 1.39Ampere

**Question 3.**

A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car?

**Answer:**

Emf of secondary cell E = 1.9Volt

Internal resistance of cell r = 380 Ω

⇒ I = 1.9V/380Ω

⇒ I = 0.005Ampere

Hence maximum current drawn from cell is 0.005Ampere and since a large current is required to start the motor of a car and hence cell cannot be used to start the motor.

**Question 4.**

Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. (ρ_{Al} = 2.63 × 10^{–8} Ωm, ρ_{Cu} = 1.72 × 10^{–8} Ωm, Relative density of Al = 2.7, of Cu = 8.9.)

**Answer:**

Resistivity of aluminium, p_{1} = 2.63 × 10^{−8} Ω m

Relative density of aluminium, d_{1} = 2.7

Let l1 be the length of aluminium wire and m1 be its mass.

Resistance of the aluminium wire = R_{1}

Area of cross-section of the aluminium wire = A_{1}

Resistivity of copper, p_{2} = 1.72 × 10^{−8} Ω metre

Relative density of copper, d_{2} = 8.9

Let l2 be the length of copper wire and m2 be its mass.

Resistance of the copper wire = R2

Area of cross-section of the copper wire = A2

We can write,

R_{1} = (p_{1} × l_{1})/A_{1}

R_{2} = (p_{2} × l_{2})/A_{2}

Since R_{1} = R_{2}

(p_{1} × l_{1})/A_{1} = (p_{2} × l_{2})/A_{2}

And l_{1} = l_{2}

Therefore p_{1}/A_{1} = p_{2}/A_{2}

A_{1}/A_{2} = (2.63 × 10-8)/(1.72 × 10^{–8})

⇒ A_{1}/A_{2} = 2.63/1.72

Mass of the aluminium wire,

m_{1} = Volume × Density

= A_{1} × l_{1} × d_{1} = A_{1} l_{1} d_{1} .....1

Mass of the copper wire, m_{2} = Volume × Density

= A_{2} × l_{2} × d_{2} = A_{2} l_{2} d_{2} .......2

Dividing equation (1) by equation (2), we obtain

m_{1}/m_{2} = (A_{1} l_{1} d_{1})/(A_{2} l_{2} d_{2})

Since l_{1} = l_{2}

m_{1}/m_{2} = A_{1} d_{1}/A_{2} d_{2}

Since A_{1}/A_{2} = 2.63/1.72 calculated above

⇒ m_{1}/m_{2} = (2.63 × 2.7)/(1.72 × 8.9)

⇒ m1/m2 = 0.46

Since m1 is smaller than m2 Hence, aluminium is lighter than copper.

Since aluminium is lighter, it is preferred for overhead power cables over copper.

**Question 5.**

What conclusion can you draw from the following observations on a resistor made of alloy manganin?

**Answer:**

From the given information shown in the table, the ratio of voltage and current is a constant value and the value is equal to 19.7 Hence manganin is a Ω ic conductor.

According to Ohm’s law the ratio of voltage and current is resistance of conductor so Resistance of manganin is 19.7 Ω .

**Question 6.**

Answer the following questions:

(a) A steady current flows in a metallic conductor of non-uniform cross-section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed?

(b) Is Ohm’s law universally applicable for all conducting elements? If not, give examples of elements which do not obey Ω ’s law.

(c) A low voltage supply from which one needs high currents must have very low internal resistance. Why?

(d) A high tension (HT) supply of, say, 6 kV must have a very large internal resistance. Why?

**Answer:**

(a) When a steady current flows in a metallic conductor of non-uniform cross section, the current flowing through the conductor is constant.

Since Current density, electric field and drift speed are inversely proportional to area of cross section therefore they are not constant.

(b) No, Ω ’s law is not applicable for all conducting elements. Example of non Ω ic conductor is Vacuum diode semiconductor.

(c) According to Ω ’s law V = I × R

If Voltage V is low, then Resistance R must be very low so that high current can be drawn from source.

(d) A high Tension supply must have a very large internal resistance so that it can limit the current from exceeding the safety limit.

**Question 7.**

Choose the correct alternative:

(a) Alloys of metals usually have (greater/less) resistivity than that of their constituent metals.

(b) Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals.

(c) The resistivity of the alloy manganin is nearly independent of/ increases rapidly with increase of temperature.

(d) The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of (10^{22}/10^{23}).

**Answer:**

(a) Alloys of metals usually have greater resistivity than that of their constituent metals.

Alloys are the homogeneous mixture of two or more metals and due to which alloys have better conducting properties as compared to pure metals. Pure metals have large number of free electrons and simple atomic arrangement but in alloys these free electrons number as well as their atomic arrangement changes. Hence alloys have greater resistivity than metals.

(b) Alloys usually have lower temperature coefficient of resistance than pure metals.

The resistance change factor per °C of temp change is called the temperature coefficient. Coefficient approaching zero can be obtained by alloying certain metals. So we can say that alloys have lower temperature coefficient of resistance than pure metals.

(c) The resistivity of alloy, manganin, is nearly independent of increase of temperature.

The resistivity of alloys is high and atoms are not arranged in proper order and with the increase in temperature resistivity increases due to collision and random motion of atoms.

(d) The resistivity of a typical insulator is greater than that of a metal by a factor of order of 10^{22}.

In case of insulator, resistivity decreases with increase of temperature.

**Question 8.**

Given n resistors each of resistance R, how will you combine them to get the (i) maximum (ii) minimum effective resistance?

What is the ratio of the maximum to minimum resistance?

**Answer:**

Total number of resistors = n

Resistance of each resistor = R

(i)

When n resistors are connected in series, effective resistance R1is the maximum,

given by the product nR.

Hence, maximum resistance of the combination, R1 = nR

(ii)

When n resistors are connected in parallel, the effective resistance (R_{2}) is the minimum, given by the ratio: R/n

Hence, minimum resistance of the combination, R_{2} = R/n

(iii) The ratio of the maximum to the minimum resistance is,

⇒

**Question 9.**

Given the resistances of 1Ω, 2Ω, 3Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω, (iii) 6 Ω, (iv) (6/11) Ω?

**Answer:**

(i)

The resistance of the given resistors is,

R1 = 1 Ω , R2 = 2 Ω , R3 = 3 Ω

Connect 2 Ω and 1 Ω resistor in parallel and 3 Ω resistor in series to them

Equivalent resistance R’ is {(2 × 1)/(2 + 1)} + 3

Hence R’ is equal to 11/3 Ω

(ii)

Now connect 2 Ω and 3 Ω resistor in parallel and one Ω resistor in series to it

Equivalent Resistance R” = {(2 × 3)/(2 + 3)} + 1

R” = 11/5 Ω

(iii)

Now connect all three resistor 1 Ω 2 Ω and 3 Ω in series

Hence total resistance R”’ is 1 + 2 + 3 = 6 Ω

(iv)

Now connect all three resistors in parallel;

Equivalent Resistance is R”” = (1 × 2 × 3)/(1 × 2 + 2 × 3 + 3 × 1)

R”” = 6/11 Ω

**Question 10.**

Determine the equivalent resistance of networks shown in Fig. 3.31.

**Answer:**

(a)

It can be observed from the given circuit that in the first small loop, two resistors

of resistance 1 Ω each are connected in series.

Hence, their equivalent resistance = (1 + 1) = 2 Ω

It can also be observed that two resistors of resistance 2 Ω each are connected in series.

Hence, their equivalent resistance = (2 + 2) = 4 Ω.

Now the given circuit diagram can be explained as

Now draw 4 loops each having 2 resistors of 2 Ω and 4 Ω connected in parallel.

Hence Equivalent Resistance R_{e} of each loop is

⇒ Re = 4/3 Ω

Since all 4 loops will be connected in series and each loop will have 4/3 Ω resistance

Hence, equivalent resistance of the given circuit is 4 × 4/3 = 16/3 Ω .

(b)

From the given circuit that five resistors of resistance R each are connected in series.

Hence, equivalent resistance of the circuit = R + R + R + R + R = 5R

**Question 11.**

Determine the current drawn from a 12V supply with internal resistance 0.5Ω by the infinite network shown in Fig. 3.32. Each resistor has 1Ω resistance.

**Answer:**

The resistance of each resistor connected in the given circuit, R = 1 Ω

Equivalent resistance of the given circuit = R’

The network is infinite. Hence, equivalent resistance is given by the relation,

R’ = 2 + (R’/R’ + 1)

or R’^{2} -2R’ -2 = 0

On solving R’ = (2+√4+8)/2

(Negative value of R’ cannot be accepted)

R’ = 1 + √3 Ω

R’ = 1 + 1.73 = 2.73 Ω

Internal resistance of the circuit, r = 0.5 Ω

Hence, total resistance of the given circuit = 2.73 + 0.5 = 3.23 Ω

Supply voltage, V = 12 V

According to Ω ’s Law, current drawn from the source is given by the ratio, = 12/3.23 = 3.72 A

**Question 12.**

Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents upto a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf e and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(a) What is the value e?

(b) What purpose does the high resistance of 600 kW have?

(c) Is the balance point affected by this high resistance?

(d) Would the method work in the above situation if the driver cell of the potentiometer had an emf of 1.0V instead of 2.0V?

(e) Would the circuit work well for determining an extremely small emf, say of the order of a few mV (such as the typical emf of a thermo-couple)? If not, how will you modify the circuit?

**Answer:**

(a) Constant emf of the given standard cell, E_{1} = 1.02 V

Balance point on the wire, l_{1} = 67.3 cm

A cell of unknown emf, ε, replaced the standard cell. Therefore, new balance point on the wire,

l = 82.3 cm

The relation connecting emf and balance point is,

E_{1}/l_{1} = E/l

E = l × E_{1}/l_{1}

E = 82.3 cm × 1.02 V/67.3 cm

E = 1.247 Volt.

The value of unknown emf is 1.247 V.

(b) The purpose of using the high resistance of 600 kΩ is to reduce the current through the galvanometer when the movable contact is far from the balance point.

(c) The balance point is not affected by the presence of high resistance.

(d) The point is not affected by the internal resistance of the driver cell.

(e) The method would not work if the driver cell of the potentiometer has an emf of 1.0 V instead of 2.0 V. This is because if the emf of the driver cell of the potentiometer is less than the emf of the other cell, then we cannot obtain balance point on the wire.

(f) The circuit would not work well for determining an extremely small emf. As the circuit would be unstable, the balance point would be close to end A. Hence, there would be a high percentage of error. The given circuit can be modified if a series resistance is connected with the wire AB. The potential drop across AB is slightly greater than the emf measured. The percentage error would be small.

**Question 13.**

Figure 3.34 shows a 2.0 V potentiometer used for the determination of internal resistance of a 1.5 V cell. The balance point of the cell in open circuit is 76.3 cm. When a resistor of 9.5Ω is used in the external circuit of the cell, the balance point shifts to 64.8 cm length of the potentiometer wire. Determine the internal resistance of the cell.

**Answer:**

Balance point for open circuit, l_{1} = 78.3 cm

Resistance of the external resistor, R = 9.5 Ω

Balance point for this resistor, l_{2} = 64.8 cm

The relation connecting internal resistance r and balance point is,

=0.177 × 9.5

=1.685 Ω

Therefore, the value of the unknown resistance, X, is 1.685 Ω.