Questions OBJECTIVE - II and Answer Electric Current in Conductors HC Verma Part II

Q#1

Electrons are emitted by a hot filament and are accelerated by an electric field as shown in the figure (32-EQ1). The two stops at the left ensure that the electron beam has a uniform cross-section. 

(a) The speed of the electron is more at B than at A.

(b) The electric current is from left to right.

(c) The magnitude of the current is larger at B than at A.

(d) The current density is more at B than at A.  

Answer: (a)   

The electrons are accelerated opposite to the electric field. Hence in the above figure, the electrons are accelerated from A to B speed. Obviously, the speed of the electron is more at B than A. Option (a) is correct.

The electric current is opposite to the flow of electrons and along the electric field. Here the current is from right to the left. Option (b) is incorrect.

Since the same number of electrons and hence charge flow through a cross-section, hence the current is the same at A and B. Options (c) and (d) are wrong.      

Q#2. 

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.

(a) There is no current through A.

(b) There is no current through B.

(c) There is a current through A as long as the charging is not complete.

(d) There is a current through B as long as the charging is not complete.     

Answer: (b), (c).    

When a capacitor is connected to a battery, negative and positive charges from the battery terminals flow through the wire to the plates of capacitors where they get stored. There is no flow of charge between the plates of the capacitor and hence no current through B. Option (b) is correct. 

Now at t =0, the plates are uncharged and hence the charges rush through the wire to the plates. So,there is a current through A until the plates are charged. Option (c) is correct.

Due to the above reason, options (a) and (d) are incorrect. 

Q#3

When no current is passed through a conductor, 

(a) the free electrons do not move

(b) the average speed of a free-electron over a large period of time is zero.

(c) the average velocity of a free electron over a large period of time is zero

(d) the average of the velocities of all the free electrons at an instant is zero.    

Answer: (c), (d).   

The free electrons in a conductor are not stationary but they move randomly and collide with metal lattice. Hence options (a) and (b) are incorrect.

But over a large period of time, the displacement of electrons is zero and hence average velocity is zero. Option (c) is correct.

When there is no current in the conductor, there is no charge transfer or collective movement of electrons in the conductor. Hence option (d) is true.

Q#4

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

(a) drift speed

(b) resistivity

(c) resistance

(d) the number of free electrons.      

Answer: (d).   

When a resistor is heated, the drift speed of electrons, resistivity of the material of the resistor, and the resistance change. Options (a), (b) and (c) are incorrect.

But the number of electrons does not change only its drift speed change. Option (d) is correct.   

Q#5. 

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

(a) increases

(b) decreases

(c) remains constant

(d) may increase or decrease depending on the actual temperature.      

Answer: (a).   

The relation between resistivity ρ and conductivity σ is 

ρ = 1/σ

Hence the ratio of resistivity to conductivity

$\frac{\rho}{\sigma}=\frac{1/ \sigma}{\sigma}=\frac{1}{\sigma^2}=\rho^2$ 

We know that with the increase in temperature, resistivity increases. Hence this ratio will also increase. Only option (a) is correct. 

Q#6. 

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

(a) the charge crossing in a given time interval

(b) drift speed

(c) current density

(d) free-electron density.        

Answer: (a), (d).   

Since the charge is not stored in a conductor, the charge crossing at a time interval at any cross-section is constant. Option (a) is correct.

Since the charge crossing a cross-section per unit time is constant, it will cross fastly at a smaller cross-section than a larger one. So the drift speed is not constant. Also, the current density is dependent on the area of cross-section. The relation is given as

$j = \frac{i}{A}=n \times e \times v \times d$

Hence options (b) and (c) are incorrect.

Free electron density is independent of the cross-section area. Option (d) is correct.

Q#7

Mark out the correct options.

(a) An ammeter should have small resistance 

(b) An ammeter should have large resistance

(c) A voltmeter should have small resistance

(d) A voltmeter should have large resistance.      

Answer: (a), (d).   

An ammeter is connected in series and a voltmeter is connected in parallel. Hence ammeter should have small resistance and the voltmeter should have large resistance so that the actual current going through the circuit and the potential difference between the given two points do not change. Option (a) and (d) are correct, options (b) and (c) are incorrect.

Q#8. 

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

(a) 5 s

(b) 50 s

(c) 500 s

(d) 500.    

Answer: All. 

When a capacitor is fully charged, the total charge on the capacitor,

Q = CV.

When the capacitor is connected in series with a resistor R, the amount of charge collected on the capacitor in initial t seconds is given as,

Q' = Q(1- e - $\frac{t}{CR}$).

Here CR = $500 \times 10^{-6} \times 10,000 \ s=5 \ s$.

So, in the initial 5 s, the charge collected on the capacitor is

Q₅ = Q(1 - e⁻¹) = 0.632Q

So, 63.20% of the total charge is collected in the initial 5 s. Since theoretically the capacitor is fully charged at t = infinity. Hence in any time period after the initial 5 s, the charge stored will be less than Q₅. So, all options are true.      

Q#9. 

A capacitor C₁ of capacitance 1 µF and a capacitor of 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.

(a) The current in each of the two discharging circuits is zero at t = 0.

(b) The currents in the two discharging circuits at t = 0 are equal but not zero.

(c) The currents in the two discharging circuits at t = 0 are unequal.

(d) C₁ loses 50% of the initial charge sooner than C₂ loses 50% of its initial charge. 

Answer: (b), (d). 

When discharging, the remaining charge on the capacitors,

$q_1=C_1Ve^{\frac{-t}{C_1R}}$, and

$q_2= C_2Ve^{\frac{-t}{C_2R}}$

The current in the first discharging circuit,

$i_1= \frac{dq_1}{dt}=C_1V  \times \frac{-1}{C_1R} \times e^{\frac{-t}{C_1R}}$

= $\frac{-V}{R}e^{\frac{-t}{C_1R}}$

Similarly the current in second circuit,

$i_2= \frac{dq_2}{dt}=C_2V  \times \frac{-1}{C_2R} \times e^{\frac{-t}{C_2R}}$

= $\frac{-V}{R}e^{\frac{-t}{C_2R}}$

Now at t = 0,

$i_1=\frac{-V}{R}$ and also $i_2=\frac{-V}{R}$.

So, the current in both discharging circuits is non-zero and equal at t = 0.

Option (b) is correct, options (a) and (c) are wrong.

Initial charge in C₁, 

$Q_1=C_1V$

At 50% loss of charge, $q_1=\frac{C_1V}{2}$.

Putting in the first discharge equation,

$\frac{C_1V}{2}=C_1Ve^{\frac{-t}{C_1R}}$

$e^{\frac{-t}{C_1R}}=\frac{1}{2}$

$\frac{et}{C_1R}=2$

$t=C_1R \ ln \ 2$

Similarly, for the second circuit, 50% discharge will take in a time 

$t'=C_2R \ ln \ 2$

So, $\frac{t}{t'}=\frac{C_1}{C_2}=\frac{1}{2}$

So C₁ loses 50% charge sooner than C₂. Option (d) is true.