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Electric Current

Moving electric charges generate electric current. Any medium in which electric charge is free to move is called a conductor of electricity. Electric charges flow from a higher potential energy state (higher electric potential) to a lower potential energy state (lower electric potential).

Positive charges flow from high potential to low potential and negative charges flow from low potential to high potential.

Electric Current (Definition) :

The charge passing through the cross-section of a conductor in unit time is called electric current.

If a charge ΔQ passes through an area in time Δt then the mean current flowing through the area in this interval of time (I_avg) is defined as :-

Average Electric Current, I_avg = ΔQ/Δt

Instantaneous electric current, $\displaystyle I=\underset{\Delta t\to 0}{\mathop{Lim}}\,\frac{\Delta Q}{\Delta t}=\frac{dQ}{dt}$

Instantaneous electric current = slope of the graph between charge and time

Unit of electric current

S.I. unit :- Ampere (A)

One Ampere :- If one coulomb of charge flows through the cross-section of a conductor in 1 second, then the current flowing will be 1 ampere.

Since 1 coulomb of charge is carried by 1/(1.6×10^-19) = 6.25×10¹⁸ electrons, therefore 1 ampere of current means the flow of 6.25 × 10¹⁸ electrons per second through any cross section of conductor.

C.G.S. unit :- biot (Bi)

Relation between ampere and biot, 1 Amp = 0.1 Bi

1 Bi = 10 Amp

A biot :- The electric current flowing in two conducting wires of negligible cross-sectional area and infinite length, placed at a distance of 1 cm in vacuum will be a biot, if a force of 2 dynes/cm is produced between them.

Note :-

Electric Current is a fundamental quantity with dimension [M⁰L⁰T⁰A¹].
It is a scalar physical quantity.
The direction of electric current is in the direction of flow of positive charge or in the direction of applied electric field. It is opposite to the direction of flow of electrons.
When current flows, the conductor remains uncharged because the charge entered per second at one end is equal to the charge released per second at the other end.
Order of free electron density in conductors = 10²⁸ electrons/m³
The value of current for a given conductor does not change with the change in cross-section.
The flow of current is also possible in vacuum, such as the flow of electrons in a television picture tube.
If a charge q moves in a circle of radius r with speed v, then time period of rotation of charge $\displaystyle T=\frac{2\pi r}{v}$ and equivalent current $\displaystyle i=\frac{q}{t}=\frac{qv}{2\pi r}$ .
If n particles each having a charge q pass per second per unit area then current associated with cross–sectional area A is given by $\displaystyle i=\frac{{{q}_{total}}}{t}=\frac{nqA}{1\sec }=nqA$
If there are n particles per unit volume each having a charge q and moving with velocity v then current through cross–sectional area A is $\displaystyle i=\frac{{{q}_{total}}}{t}=\frac{nqvA}{1\sec }=nqvA$

Example 1.

The value of charge Q passing through a surface depends on time t as follows – Q=5t³+4t+3 coulomb, then find the value of instantaneous current flowing through the surface at t = 1s.

Solution :

Instantaneous current $\displaystyle i=\frac{dq}{dt}=\frac{d}{dt}(5{{t}^{3}}+4t+3)$

$\displaystyle \Rightarrow i=15{{t}^{2}}+4$

Current flowing through the surface at t = 1s

$\displaystyle i=15{{\left( 1 \right)}^{2}}+4=15+4=19A$

Example 2.

The current flowing in a conductor is given as follows :-

i = 10 t

What will be the charge passing through the conductor in the time interval 10 seconds?

Solution :

We know that $\displaystyle i=\frac{dq}{dt}$ , So $\displaystyle dq=idt$ व $\displaystyle q=\int{idt}$

Charge passing through the conductor from t = 0 sec to t = 10 sec,

$\displaystyle q=\int\limits_{0}^{10}{idt}=\int\limits_{0}^{10}{10tdt=\left[ \frac{10{{t}^{2}}}{2} \right]}_{0}^{10}=\left[ 5{{t}^{2}} \right]_{0}^{10}$

$\displaystyle \Rightarrow q=5\left( {{10}^{2}}-{{0}^{2}} \right)$

$\displaystyle \therefore q=500C$

Example 3.

The current through a wire depends on time as i =(2 +3t)A. Calculate the charge crossed through a cross section of the wire in 10 s.

Solution :

$\displaystyle I=\frac{dq}{dt}$

$\displaystyle \Rightarrow dq=Idt=(2+3t)dt$

$\displaystyle \Rightarrow \int\limits_{0}^{Q}{dq}=\int\limits_{0}^{10}{(2+3t)dt}$

$\displaystyle \Rightarrow Q=\left[ 2t+\frac{3{{t}^{2}}}{2} \right]_{0}^{10}=\left[ 2(10)+\frac{3{{(10)}^{2}}}{2} \right]$

$\displaystyle \Rightarrow Q=20+150=170C$

Example 4.

Find the value of current from the given charge-time curve.

Solution :

We know that

Instantaneous electric current = slope of the graph between charge and time

electric current(i), Slope of q – t curve = tan θ = tan 45^o = 1 Amp.

Example 5.

The electron of a hydrogen atom travels in a circular path of radius 5.3 × 10⁻¹¹ m with a uniform speed of 2.2 × 10⁶ m/s. Find the average value of the electric current.

Solution :

$\displaystyle i=\frac{qv}{2\pi r}=\frac{1.6\times {{10}^{-19}}\times 2.2\times {{10}^{6}}}{2\pi \times 5.3\times {{10}^{-11}}}$