Suppose you have three resistors each of value 30 Ω. List all the different resistances you can obtain using them.

A proton beam is going from east to west. Is there an electric current? If yes, in what direction?

In an electrolyte, positive ions move from left to right and the negative ions from right to left. If there a net current? If yes, in what direction?

In a TV tube, the electrons are accelerated from the rear to the front. What is the direction of the current?

The drift speed is defined as v_d = Δℓ/Δt where Δℓ is the distance drifted in a long time Δt. Why don’t we define the drift speed as the limit of Δℓ/Δt as Δt → 0?

One of your friends argues that he has read in previous chapters that there can be no electric field inside a conductor. And hence there can be no current through it. What is the fallacy in this argument?

When a current is established in a wire, the free electrons drift in the direction opposite to the current. Does the number of free electrons in the wire continuously decrease?

A fan with copper winding in its motor consumes less power as compared to an otherwise similar fan having aluminum winding. Explain.

The thermal energy developed in a current-carrying resistor is given by U = i²Rt and also by U = Vit. Should we say that U is proportional to i² or to i?

Consider a circuit containing an ideal battery connected to a resistor. Do “work done by the battery” and “the thermal energy developed” represent two names of the same physical quantity?

Is work done by a battery always equal to the thermal energy developed in electrical circuits? What happens if a capacitor is connected in the circuit?

A non-ideal battery is connected to a resistor. Is work done by the battery equal to the thermal energy developed in the resistor? Does your answer change if the battery is ideal?

Sometimes it is said that “heat is developed” in a resistance when there is an electric current in it. Recall that heat is defined as the energy being transferred due to the temperature difference. Is the statement under quotes technically correct?

We often say “a current is going through the wire”. What goes through the wire, the charge or the current?

Would you prefer a voltmeter or a potentiometer to measure the emf of a battery?

Does a conductor become charged when a current is passed through it?

Can the potential difference across a battery be greater than its emf?

A metallic resistor is connected across a battery. If the number of collisions of the free electrons with the lattice is somehow decreased in the resistor (for example, by cooling it), the current will

Two resistors A and B have resistances R_A and R_B respectively with R_A < R_B. The resistivities of their materials are ρ_A and ρ_B.

The product of resistivity and conductivity of a cylindrical conductor depends on

As the temperature of a metallic resistor is increased, the product of its resistivity and conductivity.

In an electric circuit containing a battery, the charge (assumed positive) inside the battery

A resistor of resistance R is connected to an ideal battery. If the value of R is decreased, the power dissipated in the resistor will

A current passes through a resistor. Let K₁ and K₂ represent the average kinetic energy of the conduction electrons and the metal ions respectively.

Two resistors R and 2R are connected in series in an electric circuit. The thermal energy developed in R and 2R are in the ratio

Two resistances R and 2R are connected in parallel in an electric circuit. The thermal energy developed in R and 2R are in the ratio

A uniform wire of resistance 50 Ω is cut into 5 equal parts. These parts are now connected in parallel. The equivalent resistance of the combination is

Consider the following two statements:

(A) Kirchhoff’s junction law follows from conservation of charge

(B) Kirchhoff’s loop law follows from the conservative nature of electric field

Two non-ideal batteries are connected in series. Consider the following statements:

(A) The equivalent emf is larger than either of the two emfs.

(B) The equivalent internal resistance is smaller than either of the two internal resistances.

Two non-ideal batteries are connected in parallel. Consider the following statements:

(A) The equivalent emf is smaller than either of the two emfs.

(B) The equivalent internal resistance is smaller than either of the two internal resistances.

The net resistance of an ammeter should be small to ensure that

The net resistance of a voltmeter should be large to ensure that

Consider a capacitor-charging circuit. Let Q₁ be the charge given to the capacitor in a time interval of 10 ms and Q₂ be the charge given in the next time interval of 10ms. Let 10μC charge be deposited in a time interval t₁ and the next 10 μC charge is deposited in the next time interval t₂.

Electrons are emitted by a hot filament and are accelerated by an electric field as shown in figure. The two stops at the left ensure that the electrons beam has a uniform cross-section.

A capacitor with no dielectric is connected to a battery at t = 0. Consider a point A in the connecting wires and a point B in between the plates.

When no current is passed through a conductor.

Which of the following quantities do not change when a resistor connected to a battery is heated due to the current?

As the temperature of a conductor increases, its resistivity and conductivity change. The ratio of resistivity to conductivity.

A current passes through a wire of non-uniform cross-section. Which of the following quantities are independent of the cross-section?

Mark out the current options.

A capacitor of capacitance 500 μF is connected to a battery through a 10 kΩ resistor. The charge stored on the capacitor in the first 5s is larger than the charge stored in the next.

A capacitor C₁ of capacitance 1 μF and a capacitor C₂ of capacitance 2 μF are separately charged by a common battery for a long time. The two capacitors are then separately discharged through equal resistors. Both the discharge circuits are connected at t = 0.

The amount of charge passed in time t through a cross-section of a wire is Q(t) = At² + Bt + C.

(a) Write the dimensional formulae for A, B and C.

(b) If the numerical values of A, B and C are 5, 3 and 1 respectively in S.I. units, find the value of the current at t = 5 s.

An electron gun emits 2.0 × 10¹⁶ electrons per second. What electric current does this correspond to?

The electric current existing in a discharge tube is 2.0 μA. How much charge is transferred across a cross-section of the tube in 5 minutes?

The current through a wire depends on time as

i = i₀ + αt,

Where i₀ = 10 A and α = 4 As^–1. Find the charge crossed through a section of the wire in 10 seconds.

A current of 1.0 A exists in a copper wire of cross-section 1.0 mm². Assuming one free electron per atom calculate the drift speed of the free electrons in the wire. The density of copper is 9000 kg m^–3.

A wire of length 1 m and radius 0.1 mm has a resistance of 100 Ω. Find the resistivity of the material.

A uniform wire of resistance 100 Ω is melted and recast in a wire of length double that of the original. What would be the resistance of the wire?

Consider a wire of length 4 m and cross-sectional area 1 mm² carrying a current of 2 A. If each cubic meter of the material contains 10²⁹ free electrons, find the average time taken by an electron to cross the length of the wire.

What length of a copper wire of cross-section area 0.01 mm² will be needed to prepare a resistance of 1 kΩ? Resistivity of copper = 1.7 × 10^–8 Ω m.

Figure shows conductor of length ℓ having a circular cross-section. The radius of cross-section varies linearly from a to b. The resistivity of the material is ρ. Assuming that b – a << ℓ, find the resistance of the conductor.

A copper wire of radius 0.1 mm and resistance 1 kΩ is connected across a power supply of 20 V.

(a) How many electrons are transferred per second between the supply and the wire at one end?

(b) Write down the current density in the wire.

Calculate the electric field in a copper wire of cross-section area 2.0 mm² carrying a current of 1 A. The resistivity of copper = 1.7 × 10^–8 Ω m.

A wire has a length of 2.0 m and a resistance of 5.0 Ω. Find the electric field existing inside the wire if it carries a current of 10 A.

The resistances of an iron wire and a copper wire at 20°C and 3.9 Ω and 4.1 Ω respectively. At what temperature will the resistances be equal? Temperature coefficient of resistivity for iron is 5.0 × 10^–3 K^–1 and for copper it is 4.0 × 10^–3 K^–1. Neglect any thermal expansion.

The current in a conductor and the potential difference across its ends are measured by an ammeter and a voltmeter. The meters draw negligible currents. The ammeter is accurate but the voltmeter has a zero error (that is, it does not read zero when no potential difference is applied). Calculate the zero error if the readings for two different conditions are 1.75 A, 14.4 V and 2.75 A, 22.4 V.

Figure shows an arrangement to measure the emf ϵ and internal resistance r of a battery. The voltmeter has a very high resistance and the ammeter also has some resistance. The voltmeter reads 1.52 V when the switch S is open. When the switch is closed the voltmeter reading drops to 1.45 V and the ammeter reads 1.0 A. Find the emf and the internal resistance of the battery.

The potential difference between the terminals of a battery of emf 6.0 V and internal resistance 1 Ω drops to 5.8 V when connected across an external resistor. Find the resistance of the external resistor.

The potential difference between the terminals of a 6.0 V battery is 7.2 V when it is being charged by a current of 2.0 A. What is the internal resistance of the battery?

The internal resistance of an accumulator battery of emf 6 V is 10 Ω when it is fully discharged. As the battery gets charged up, its internal resistance decreases to 1Ω. The battery in its completely discharged state is connected to a charger which maintains a constant potential difference of 9V. Find the current through the battery

(a) just after the connections are made and

(b) after a long time when it is completely charged.

Find the value of i₁/i₂ in figure if (a) R = 0.1Ω, (b) R = 1 Ω (c) R = 10 Ω. Note from our answers that in order to get more current from a combination of two batteries they should be joined in parallel if the external resistance is small and in series if the external resistance is large as compared to the internal resistances.

Consider N = n₁n₂ identical cells, each of emf ϵ and internal resistance r. Suppose n₁ cell are joined in series to form a line and n₂ such lines are connected in parallel. The combination drives a current in an external resistance R.

(a) Find the current in the external resistance.

(b) Assuming that n₁ and n₂ can be continuously varied, find the relation between n₁, n₂, R and r for which the current in R is maximum.

A battery of emf 100 V and a resistor of resistance 10 kΩ are joined in series. This system is used as a source to supply current to an external resistance R. If R is not greater than 100 Ω, the current through it is constant up to two significant digits. Find its value. This is the basic principle of a constant-current source.

If the reading of ammeter A₁ in figure is 2.4A, what will the ammeters A₂ and A₃ read? Neglect the resistances of the ammeters.

The resistance of the rheostat shown in figure is 30 Ω. Neglecting the meter resistance, find the minimum and maximum currents through the ammeter as the rheostat is varied.

Three bulbs, each having a resistance of 180 Ω, are connected in parallel to an ideal battery of emf 60 V. Find the current delivered by the battery when

(a) all the bulbs are switched on,

(b) two of the bulbs are switched on and

(c) only one bulb is switched on.

Suppose you have three resistors of 20Ω, 50Ω and 100Ω. What minimum and maximum resistances can you obtain from these resistors?

A bulb is made using two filaments. A switch selects whether the filaments are used individually or in parallel. When used with a 15 V battery, the bulb can be operated at 5W, 10W or 15W. What should be the resistances of the filaments?

Figure shows a part of a circuit. If a current of 12 mA exists in the 5 kΩ resistor, find the currents in the other three resistors. What is the potential difference between the points A and B?

An ideal battery sends a current of 5A in a resistor. When another resistor of value 10 Ω is connected in parallel, the current through the battery is increased to 6A. Find the resistance of the first resistor.

Find the equivalent resistance of the network shown in figure between the points a and b.

A wire of resistance 15.0 Ω is bent to form a regular hexagon ABCDEFA. Find the equivalent resistance of the loop between the points A and B, B A and C and C A and D.

Consider the circuit shown in figure. Find the current through the 10Ω resistor when the switch S is A. open B. closed.

Find the currents through the three resistors shown in figure.

Figure shows a part of an electric circuit. The potentials at the points a, b and c are 30 V, 12 V and 2 V respectively. Find the currents through the three resistors.

Each of the resistors shown in figure has a resistance of 10 Ω and each of the batteries has an emf of 10 V. Find the currents through the resistors a and b in the two circuits.

Find the potential difference V_a – V_b in the circuits shown in figure.

In the circuit shown in figure ϵ₁ = 3V, ϵ₂ = 2V, ϵ₃ = 1V and r₁ = r₂ = r₃ = 1Ω. Find the potential difference between the points A and B and the current through each branch.

Find the current through the 10Ω resistor shown in figure.

Find the current in the three resistors shown in figure.

What should be the value of R in figure for which the current in it is zero.

Find the equivalent resistance of the circuits shown in figure between the points a and b. Each resistor has a resistance r.

Find the current measured by the ammeter in the circuit shown in figure.

Consider the circuit shown in figure. Find (a) the current in the circuit, (b) the potential drop across the 5Ω resistor, (c) the potential drop across the 10Ω resistor. (d) Answer the parts (a), (b) and (c) with reference to figure.

Twelve wires, each having equal resistance r, are joined to form a cube as shown in figure. Find the equivalent resistance between the diagonally opposite points a and f.

Concept/Formula used:

Kirchhoff’s junction rule:

The sum of currents entering a junction is equal to the sum of currents leaving it.

Kirchhoff’s loop rule:

The sum of potential differences around a closed loop is zero.

Find the equivalent resistances of the networks shown in figure between the points a and b.

An infinite ladder is constructed with 1Ω and 2Ω resistors as shown in figure.

(a) Find the effective resistance between the points A and B.

(b) Find the current that passes through the 2Ω resistor nearest to the battery.

The emf ϵ and the internal resistance r of the battery shown in figure are 4.3 V and 1.0 Ω respectively. The external resistance R is 50Ω. The resistances of the ammeter and voltmeter are 2.0Ω and 200Ω respectively.

(a) Find the readings of the two meters.

(b) The switch is thrown to the other side. What will be the readings of the two meters now?

A voltmeter of resistance 400 Ω is used to measure the potential difference across the 100Ω resistor in the circuit shown in figure.

(a) What will be the reading of the voltmeter?

(b) What was the potential difference across 100Ω before the voltmeter was connected?

The voltmeter shown in figure reads 18 V across the 50 Ω resistor. Find the resistance of the voltmeter.

A voltmeter consists of a 25Ω coil connected in series with a 575Ω resistor. The coil takes 10 mA for full scale deflection. What maximum potential difference can be measured on this voltmeter?

An ammeter is to be constructed which can read currents up to 2.0 A. If the coil has a resistance of 25Ω and takes 1 mA for full-scale deflection, what should be the resistance of the shunt used?

A voltmeter coil has resistance 50.0Ω and a resistor of 1.15 kΩ is connected in series. It can read potential differences upto 12 volts. If this same coil is used to construct an ammeter which can measure currents up to 2.0 A, what should be the resistance of the shunt used?

The potentiometer wire AB shown in figure is 40 cm long. Where should the free end of the galvanometer be connected on AB so that the galvanometer may show zero deflection?

The potentiometer wire AB shown in figure is 50 cm long. When AD = 30 cm, no deflection occurs in the galvanometer. Find R.

A 6-volt battery of negligible internal resistance is connected across a uniform wire AB of length 100 cm. The positive terminal of another battery of emf 4V and internal resistance 1Ω is joined to the point A as shown in figure. Take the potential at B to be zero.

(a) What are the potentials at the points A and C?

(b) At which point D of the wire AB, the potential is equal to the potential at C?

(c) If the points C and D are connected by a wire, what will be the current through it?

(d) If the 4V battery is replaced by 7.5 V battery, what would be the answers of parts (a) and (b)?

Concepts/Formula used:

Ohm’s Law:

Potential Difference (V) across a resistor of resistance R when current I passes through it is given by Ohm’s law:

Let the area of cross section of the wire (resistor) be A and resistivity be ρ.

Then,

Consider the potentiometer circuit arranged as in figure. The potentiometer wire is 600 cm long.

(a) At what distance from the point A should the jockey touch the wire to get zero deflection in the galvanometer?

(b) If the jockey touches the wire at a distance of 560 cm from A, what will be the current in the galvanometer?

Find the charge on the capacitor shown in figure.

(a) Find the current in the 20 Ω resistor shown in figure.

(b) If a capacitor of capacitance 4 μF is joined between the points A and B, what would be the electrostatic energy stored in it in steady state?

Find the charges on the four capacitors of capacitances 1 μF, 2 μF, 3 μF and 4 μF shown in figure.

Find the potential difference between the points A and B between the points B and C of figure in steady state.

A capacitance C, a resistance R and an emf ϵ are connected in series at t = 0. What is the maximum value of

(a) the potential difference across the resistor,

(b) the current in the circuit,

(c) the potential difference across the capacitor,

(d) the energy stored in the capacitor,

(e) the power delivered by the battery and

(f) the power converted into heat.

A parallel-plate capacitor with plate area 20 cm² and plate separation 1.0 mm is connected to a battery. The resistance of the circuit is 10 kΩ. Find the time constant of the circuit.

A capacitor of capacitance 10 μF is connected to a battery of emf 2V. It is found that it takes 50 ms for the charge on the capacitor to become 12.6 μC. Find the resistance of the circuit.

A 20 μF capacitor is joined to a battery of emf 6.0 V through a resistance of 100 Ω. Find the charge on the capacitor 2.0 ms after the connections are made.

The plates of a capacitor of capacitance 10 μF, charged to 60 μC, are joined together by a wire of resistance 10Ω at t = 0. Find the charge on the capacitor in the circuit at

(a) t = 0, (b) t = 30 μs, (c) t = 120 μs and (d) t = 1.0 ms.

A capacitor of capacitance 8.0 μF is connected to a battery of emf 6.0 V through a resistance of 24Ω. Find the current in the circuit

(a) just after the connections are made and

(b) one time constant after the connections are made.

A parallel-plate capacitor of plate area 40 cm² and separation between the plates 0.10 mm is connected to a battery of emf 2.0 V through a 16 Ω resistor. Find the electric field in the capacitor 10 ns after the connections are made.

A parallel-plate capacitor has plate area 20 cm², plate separation 1.0 mm and a dielectric slab of dielectric constant 5.0 filling up the space between the plates. This capacitor is joined to a battery of emf 6.0 V through a 100 kΩ resistor. Find the energy for the capacitor 8.9 μs after the connections are made.

A 100 μF capacitor is joined to a 24 V battery through a 1.0 MΩ resistor. Plot qualitative graphs

(a) between current and time for the first 10 minutes and

(b) between charge and time for the same period.

How many time constants will elapse before the current in a charging RC circuit drops to half of its initial value? Answer the same question for a discharging RC circuit.

How many time constants will elapse before the charge on a capacitor falls to 0.1% of its maximum value in a discharging RC circuit?

How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?

How many time constants will elapse before the power delivered by the battery drops to half of its maximum value in an RC circuit?

A capacitor of capacitance C is connected to a battery of emf ϵ at t = 0 through a resistance R. Find the maximum rate at which energy is stored in the capacitor. When does the rate has this maximum value?

A capacitor of capacitance 12.0 μF is connected to a battery of emf 6.00 V and internal resistance 1.00 Ω through resistanceless leads 12.0 μs after the connections are made, what will be

(a) the current in the circuit,

(b) the power delivered by the battery,

(c) the power dissipated in heat and

(d) the rate at which the energy stored in the capacitor is increasing.

A capacitance C charged to a potential difference V is discharged by connecting its plates through a resistance R. Find the heat dissipated in one time constant after the connections are made. Do this by calculating ∫i₂R dt and also by finding the decrease in the energy stored in the capacitor.

By evaluating ∫i²R dt, show that when a capacitor is charged by connecting it to a battery through a resistor, the energy dissipated as heat equals the energy stored in the capacitor.

A parallel-plate capacitor is filled with a dielectric material having resistivity ρ and dielectric constant K.

The capacitor is charge and disconnected from the charging source. The capacitor is slowly discharged through the dielectric. Show that the time constant of the discharge is independent of all geometrical parameters like the plate area or separation between the plates. Find this time constant.

Find the charge on each of the capacitors 0.20 ms after the switch S is closed in figure.

The switch S shown in figure is kept closed for a long time and is then opened at t = 0. Find the current in the middle 10 Ω resistor at t = 1.0 ms.

Concepts/Formulas used:

When is capacitor is discharging through as resistor of resistance R, the current through the resistor is given by:

where I₀ is the initial current and C is the capacitance of the capacitor.

A capacitor of capacitance 100 μF is connected across a battery of emf 6.0 V through a resistance of 20 kΩ for 4.0 s. The battery is then replaced by a thick wire. What will be the charge on the capacitor 4.0 s after the battery is disconnected?

Consider the situation shown in figure. The switch is closed at t = 0 when the capacitors are uncharged. Find the charge on the capacitor C₁ as a function of time t.

A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an uncharged capacitor of equal capacitance through a resistance R. Find the charge on the second capacitor as a function of time.

A capacitor of capacitance C is given a charge Q. At t = 0, it is connected to an ideal battery of emf ϵ through a resistance R. Find the charge on the capacitor at time t.