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Electric Current

The electric current is given by:

I = V / R

Corresponding units:
ampere (A) = volt (V) / ohm (Ω)

This formula is derived from Ohm's law. Where we have:
V: voltage
I: current
R: resistance

If the electric power and the total resistance are known, then the current can be determined by using the following formula:


I = √(P / R)

Corresponding units:
Ampere (A) = √(Watt (W) / Ohm (Ω))

Where P is the electric power.



Electric Current
The rate of flow of charge through a cross section of some region of a metallic wire (or an electrolyte) is called the current through that region.

If rate of flow of charge is not constant then the current at any instant is given by the differential limit: I = dQ/dt.

If a charge Q flows through the circuit for time t, then

  I = Q/t.

The S.I unit of current is called ampere (A) (coulomb/second).
1 ampere = 6.25 × 10
8 electrons/ sec

In metallic conductors the current is due to the motion electrons whereas in electrolytes and ionized gases, both electrons and positive ions move in opposite direction. The direction of current is taken as the direction in which positive charges move.

In conduction although the current is only due to electrons, the current was earlier assumed to be due to positive charges flowing from the positive of the battery to the negative. The direction of current therefore is taken as opposite to the flow of electrons.



If current is constant:  Δq = I.Δt 

function of time:



Charge= Area under the graph = ½ × t0 × I0

To Find Current in Electrical Circuit
For Simple circuit or single wire, we have:


For complex circuit with more than one wire we can determine current by the means of Kirchhoff’s two laws

First Law: This law is based on the principle of conservation of charge and states that in an electrical circuit (or network of wires) the algebraic sum of currents meeting at a point is zero. 


The arrow head marked in circuit represents the direction of conventional current i.e. the direction of flow of positive charge, whereas the direction of flow of electrons gives the direction of electronic current which is opposite to that of conventional current.
I1 + I4 + I5 = I3 + I2 + I6


Second Law: The algebraic sum of the product of the current and resistance in any closed loop of a circuit is equal to the algebraic sum of electromotive forces acting in that loop.
Mathematically .


Electromotive forces – The emf (𝜖) of the source is defined as the work done per unit charge in taking a positive charge through the seat of emf from the low potential end to the high potential end. Thus,
𝜖 = w/Q

When no current flows, the
emf of the source is exactly equal to the potential difference between its ends. The unit of emf is the same as that of potential  i.e. volt.

The average flow of electrons in a conductor not connected to battery is zero i.e. the number of free electrons crossing any section of the conductor from left to right is equal to the number of electrons crossing the section of the conductor right to left .Thus no current flows through the conductor until it is connected to the battery.


Drift velocity of free electrons in a metallic conductor 


In the absence of an electric field, the free electrons in a metal randomly in all directions and therefore their average velocity are zero. When an electric field is applied, they are accelerated opposite to the direction of the field and therefore they have a net drift in that direction. However, due to frequent collisions with the atoms, their average velocity is very small. This average velocity with which the electrons move in a conductor under a potential difference is called the drift velocity.

If E is the applied field ,e is the charge of an electron, m is the mass of an electron and τ is the time interval between successive collisions(relaxation time) ,then the acceleration of the electron is  



Since the average velocity just after a collision is zero and just before the next collision, it is a τ, the drift velocity must be:


If I is the current through the conductor and n is number of free electrons per unit volume, then it can be shown that:

The
mobility  µ of a charge carrier is defined as the drift velocity per unit electric field:


Current density (J)
(i) 
(ii)  S.I Unit of J = Am-2.
(iii)  Current density is a vector quantity its direction is that of the flow of positive charge at the given point inside the conductor.
(iv)  Dimensions of current density =
[M0L-2ToA1]

Current carriers: The charged particles whose flow in a definite direction constitutes the electric current are current carriers. The current carriers can have positive or negative charge. The Current is carried by electrons in conductors, ions in electrolytes and electrons and holes in semi conductors.


Example 1: A particle having charge q coulomb describes a circular orbit. If radius of the orbit is R and frequency of the orbital motion of particles is f, then find the current in the orbit.
 

Solution: Through any section of the orbit, the charge passes f times in one second. Therefore, through that section, total charge passing in one second is fq. By definition i = fq.


Example 2: The current in a wire varies with time according to the equation I = 4 + 2t, where I is in ampere and t is in sec. Calculate the quantity of charge which has passed through a cross-section of the wire during the time t = 2 sec to t = 6 sec.  

Solution: Let dq be the change which has passed in a small interval of time dt.
Then dq = I dt = (4+2t)dt

Hence, total charge passed during the interval t = 2 sec and t = 6 is

q = ∫62 (4 + 2t) dt = 48 coulomb

Example 3: Given a current carrying wire of non-uniform cross-section. Which of the following is constant throughout the wire?
(a) Current only
(b) Current and drift Speed
(c) Drift speed only
(d) Current, drift speed

Solution:(a)

Example4: When potential difference across a given copper wire is increase, drift velocity of
charge carriers:

(a) Decreases
(b) Increases
(c) Remain same
(d) Get reduced to zero
Solution :(b)



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