Informative line

### Current

Learn current definition in physics and relation between drift velocity and current. Practice direction of current in the given conductor through an area and instantaneous and average current.

# Introduction to Current

Current is defined as the motion of charges.

## Net Current in an Isolated Conductor

• Consider an isolated metallic conductor in which electrons are moving randomly.
• The rate of flow of electrons from left to right at any point P is same as the rate of flow of electrons from right to left due to random motion of electrons.
• So, net rate of flow of current is zero.  ## Direction of Current

• The direction of current is in the direction of flow of positive charge and opposite to the direction of flow of negative charge.  #### The direction of current in the given conductor through an area A is

A Right to left

B Left to right

C Upward

D downward

×

Direction of current is in the direction of flow of positive charge.

Hence, option B is correct.

### The direction of current in the given conductor through an area A is A

Right to left

.

B

Left to right

C

Upward

D

downward

Option B is Correct

# Instantaneous and Average Current

• Consider a metallic wire of area A through which charged particles are flowing.  • Let  $$\Delta Q$$  be the amount of charge passing through area A in  $$\Delta t$$ time interval.
• The average current through this area is the rate of flow of charges passing through it.

$$I_{avg}=\dfrac {\Delta Q}{\Delta t}$$

and instantaneous current

$$I= \displaystyle\lim_{\Delta t \to 0} \dfrac {\Delta Q}{\Delta t}=\dfrac {dQ}{dt}$$

#### Determine the value of current through a conductor of area A through which the charge is flowing as a function of time, Q = 3t2 + 2t at t = 1 sec.

A 8 A

B 10 A

C 16 A

D 25 A

×

Given :  Q = 3t2 + 2t , t = 1 sec

The average current through this area is the rate of flow of charges passing through it.

$$I_{avg}=\dfrac {dQ}{dt}$$

$$I=\dfrac {d}{dt}(3t^2+2t)$$

$$I=6t+2$$

$$(I)_{t=1{\text{ sec}}}=6×1+2$$

$$(I)_{t=1{\text{ sec}}}=8 \,A$$

### Determine the value of current through a conductor of area A through which the charge is flowing as a function of time, Q = 3t2 + 2t at t = 1 sec.

A

8 A

.

B

10 A

C

16 A

D

25 A

Option A is Correct

# Flow of Charge in an Exponential / Logarithmic Function

Differentiation of Exponential / Logarithmic Function

$$\dfrac {d}{dt}e^t=e^t; \ \\\dfrac {d}{dt}ln\ t=\dfrac {1}{t}$$

#### Consider, a charge is flowing through a conductor as a function of time Q = 3e2t, calculate the value of current at time t = 1 sec.  [ e2 = 7.389 ]

A 44.33 A

B 1 A

C 0.90 A

D 1.5 A

×

$$I=\dfrac {dQ}{dt}$$

$$I=\dfrac {dQ(t)}{dt}$$

Given : Q = 3e2t ,  t = 1 sec

$$I(t)=\dfrac {d}{dt}(3e^{2t})$$

or,  $$I(t)=3\;e^{2t}\;×(2)$$

or, $$I(t)=6e^{2t}$$

or, $$I(t)=6e^{2}$$          $$(\because t=1\ sec)$$

or, $$I(t)=6×7.389$$

or, $$I(t) = 44.33 A$$

### Consider, a charge is flowing through a conductor as a function of time Q = 3e2t, calculate the value of current at time t = 1 sec.  [ e2 = 7.389 ]

A

44.33 A

.

B

1 A

C

0.90 A

D

1.5 A

Option A is Correct

# Average Current

• Consider, a charge 'q' rotating in a circle attached at the end of an insulating string of length 'r' with constant angular velocity $$\omega$$. It is equivalent to current carrying loop.

$$I_{eq}=\dfrac {q}{T}$$  • Time period of revolution, $$T=\dfrac {2\pi}{\omega}$$

$$I_{avg}=\dfrac {q}{\dfrac {2\pi}{\omega}}$$

$$I_{avg}=\dfrac {\omega\,q}{2\pi}{}$$

#### A combination of charges $$q_1$$ and $$q_2$$ are revolving, as shown in figure. Find the correct expression of current for this loop.

A $$\dfrac {\omega}{\pi}(q_1+q_2)$$

B $$\omega(q_1+q_2)$$

C $$\dfrac {\omega}{2\pi}(q_1+q_2)$$

D $$\dfrac {\omega}{\pi}(q_1-q_2)$$

×

Time period of revolution,

$$T=\dfrac {2\,\pi}{\omega}$$

$$I_1=\dfrac {\omega\, q_1}{2\,\pi}$$

$$I_2=\dfrac {\omega\, q_2}{2\,\pi}$$ $$I=I_1+I_2$$

$$I=\dfrac {\omega}{2\pi}(q_1+q_2)$$ ### A combination of charges $$q_1$$ and $$q_2$$ are revolving, as shown in figure. Find the correct expression of current for this loop. A

$$\dfrac {\omega}{\pi}(q_1+q_2)$$

.

B

$$\omega(q_1+q_2)$$

C

$$\dfrac {\omega}{2\pi}(q_1+q_2)$$

D

$$\dfrac {\omega}{\pi}(q_1-q_2)$$

Option C is Correct

# Drift Velocity

• Drift velocity of charges can be understood from the concept of river-boat.
• Consider a situation of river boat in which boat has to reach from point P to A. But due to flow of river, the boat reaches at some point B.
• The distance between point A and B is known as drift.  • Similarly in conductors in the absence of an electric field, electrons move randomly.
• When electric field is applied, electrons drift in some direction due to force on electrons. This direction is in the direction of force means opposite to the direction of electric field.    #### A conductor is placed in an electric field $$\vec E$$, as shown in figure. What will be the direction of drift velocity of electron?

A In the same direction of electric field

B In the opposite direction of electric field

C Perpendicular direction to electric field

D None of these

×

The direction of drift velocity of free electron is opposite to the direction of the electric field. ### A conductor is placed in an electric field $$\vec E$$, as shown in figure. What will be the direction of drift velocity of electron? A

In the same direction of electric field

.

B

In the opposite direction of electric field

C

Perpendicular direction to electric field

D

None of these

Option B is Correct

# Relation between Drift Velocity and Current

• Consider a conductor of cross - sectional area A. Let "e" be the charge on each electron and "n" is the number of free electron per unit volume.  • Volume of conductor = $$A\,v_d\;\Delta t$$
• Charge crossing this area in  $$\Delta t$$  time

$$\Delta Q=n\,A\,v_d\,\Delta t\,e$$

$$i=\dfrac {\Delta Q}{\Delta t}$$

$$i = n\,A\,v_d\,e$$

#### Consider a conductor of cross sectional area of "A" through which current $$I$$  is passing. If area of cross-section is made A/2, which parameter will change accordingly?

A n gets halved

B $$v_d$$ gets tripled

C Current gets halved

D current gets doubled

×

$$I=n\,A\,v_d\,e$$

$$I\propto A$$

So, only the value of current depends on area. When area is halved, current also gets halved.

### Consider a conductor of cross sectional area of "A" through which current $$I$$  is passing. If area of cross-section is made A/2, which parameter will change accordingly?

A

n gets halved

.

B

$$v_d$$ gets tripled

C

Current gets halved

D

current gets doubled

Option C is Correct

# Calculation of Charge

• Consider a current flowing through a conductor as a function of time $$I$$ = f (t)

### Flow of Charge in a given Time Interval ($$\Delta t$$)

We know,

$$I=\dfrac {dQ}{dt}$$

$$f(t)=\dfrac {dQ}{dt}$$

or, $$dQ=f(t)\, dt$$

or, $$\int\limits_0^QdQ=\int\limits_0^t\,f(t) \,dt$$

$$Q(t)=\int\limits_0^t f(t)\,dt$$

#### Consider a current flowing through a conductor is given by $$I$$ = 3t2 + 2. Find the amount of charge flow in the time interval t = 0 to t = 2 sec.

A 11 C

B 13 C

C 12 C

D 14 C

×

$$I=\dfrac {dQ}{dt}$$

$$dQ=I\cdot dt$$

$$\int\limits_0^QdQ=\int\limits_0^t\,I\;dt$$

$$Q=\int_0^2(3t^2+2).dt$$

or, $$Q=\left [ \dfrac {3t^3}{3}+2t \right ]_0^2$$

or, Q = (2)3 + 2 × 2

Q = 12 C

### Consider a current flowing through a conductor is given by $$I$$ = 3t2 + 2. Find the amount of charge flow in the time interval t = 0 to t = 2 sec.

A

11 C

.

B

13 C

C

12 C

D

14 C

Option C is Correct

# Calculation of Drift Velocity

The relation between drift velocity and current is given by

$$i =nAv_de$$

#### Calculate the drift velocity of electron when $$I= 1 \,A$$  of current flow through a conductor of cross sectional area A = 2 m2. The number of free electrons in 1 m3, are n = 8.5 × 1022 /m3.

A 50 $$\mu$$ m/sec

B 60 $$\mu$$ m/sec

C 39 $$\mu$$m/sec

D 36 $$\mu$$ m/sec

×

We know

$$i = nev_dA$$

$$v_d = \dfrac{i}{neA}$$

$$v_d = \dfrac{1}{8.5 × 10^ {22} × 1.6 × 10 ^{-19} × 2}$$

$$v_d$$ = 0.036 × 10–3

$$v_d$$ = 36 $$\mu$$m/sec

### Calculate the drift velocity of electron when $$I= 1 \,A$$  of current flow through a conductor of cross sectional area A = 2 m2. The number of free electrons in 1 m3, are n = 8.5 × 1022 /m3.

A

50 $$\mu$$ m/sec

.

B

60 $$\mu$$ m/sec

C

39 $$\mu$$m/sec

D

36 $$\mu$$ m/sec

Option D is Correct