Electric Current and Current Density

- Electric current is the motion of electrically charged particles from one region to another. This motion of electric charges takes place within electric circuit which is a conducting path forming a closed loop.
- In a conductor electric charge will flow from its one end to another if and only if both the ends of the conductor are at different electric potentials.
- Continous flow of electric current in a conductor for a relatively long period of time can be attained using battries which could supply continous flow of charge at low potentials.

Electric current
- Here in this section we would discuss about the electric current in conductors.
- When electric field inside the conductor is zero then there would be no net flow of current in the conductor because in this case electrons in the conductors moves about randomly leaving no net flow of charge in any direction and without the flow of charge there would be no net electric current.
-To maintain a constant current in a conductor we would have to ensure that a constant and steady electric field is established inside the conductor in order to maintain a force on the mobile charges in the conductor.
-Once the electric field is maintained inside the conductor charged particles in the conductors are now under the influence of driving force F = qE.
- In an conductor charged particles undergoes frequent inelastic collision with fixed massive ions in the conductor and undergoes random change in the direction of motion.
- Hence on an average charged particle moves in the direction of driving force with an average velocity known ad drift velocity.
- Electric current is defined as the quantity of charge ΔQ flowing through cross-sectional area A in time interval Δt . Thus,
Iav = ΔQ/Δt
which is the average current flow in time Δt.
- If dQ is the amount of charge flowing in infinitesimally time interval dt through a cross-sectional area of the conductor then instantaneous current I is defined as
I = dQ/dt
- Electric current is a scalar quantity.
- SI unit of current is Ampere (A) where 1A is one coulomb per second.

Drift velocity and Current density - Consider a portion of a conductor of cross-sectional area A. Also consider a small section of conductor of length Δx. Now volume of conductor under consideration is AΔx.
- If there are n number of mobile charge carriers per unit volume then total charge in the section under consideration is
ΔQ = (number of charge carriers in the section) x (charge per carrier)
= (nAΔx)q .................1
where q is the amout of charge on each carrier.
- If charge carriers are moving with speed vd, then distance travelled by the charge carriers is Δx = vdΔt. Putting this in equation 1 we have,
ΔQ = (nAvdΔt)q
- Now current in the conductor is
I = ΔQ/Δt
thus,
I = nqvdA ....................2
where vd is average velocity known as drift velocity as defined earlier.
- Current density j is defined as current per unit cross-sectional area. Thus from 2
j = I/A = nqvd
Current is a scalar quantity but current density can also be defined too include both magnitude and direction. Thus vector current density is
j = nqvd
Direction of current density is same as the direction of electric field.
- Unit of current density is A.m-2.
- Current density tells us about how charges flow at a certain point and also about the direction of the flow at that point but current describes how charges flow throughout an extended object.

Solved example
In this solved example we will lern to apply the principle of current to the problems.

Question:-Amount of charge that passes through a certain conducting wire in 4 sec is 6.5 C. Find (a) the current in the wire and (b) number of electrons that passes through the wire in 8 sec.
Solution : -
(a)
Problem solving strategy
1. From the lesson learned about electric current identify the equation for calculating current when charge and time are given.
2. Put the values of charge and time at respective place and calculate the answer.

In this case Q = 6.5 C and t = 4 sec.
now I = q/t = 6.5C/4 sec = 1.625 A
which is the required answer.

(b)
Problem solving strategy
1. Here we have to find total charge through the wire in a given time.
2. Total charge through the wire would be equal to number of electrons through the circuit multiplied by the charge on each electron.

Here we have to find the number of electrons passing through the wire in 8 sec. If qe is the amount of charge on a single electron and total n number of electrons passes through the wire then
I = nqe/t
or, N = It/qe
putting values of I , t and qe = 1.6 x 10-19 and calculating we get
N = 8.125 x 1019

Same way problems related to current density and drift velocity can be solved.

Vector Algebra 1

 Here in this post we will go through a quick recap of vector algebra keeping in mind that reader already had detail knowledge and problem solving skills related to the topic being discussed. Here we are briefing Vector Algebra because concepts of electrostatics , electromagnetism and many more physical phenomenon can best be conveniently expressed using this tool.

A vector is a quantity that requires both a magnitude (= 0) and a direction in space it can be represented by an arrow in space for example electrostatic force, electrostatic field etc. In symbolic form we will represent vectors by bold letters. In component form vector A is written as
A = Axi+ Ayj+Azk


ADDITION OF VECTORS
Two vectors A and B can be added together to give another resultant vector C.
C = A + B

SUBTRACTION OF VECTORS
Two vectors A and B can be subtracted to give another resultant vector D.
D = A - B = A + (-B)

SCALAR MULTIPLICATION OF VECTOR
When we multiply any vector A with any scalar quantity 'n' then it's direction remains unchanged and magnitude gets multiplied by 'n'. Thus,
n(A) = nA
Scalar multiplication of vectors is distributive i.e.,
n(A + B) = nA +nB

DOT PRODUCT OF VECTORS
Dot product of two vectors A and B is defined as the product of the magnitudes of vectors A and B and the cosine of the angle between them when both te vectors are placed tail to tail. Dot product is represented as A.B thus,
A.B = |A| |B| cosθ
where θ is the angle between two vectors.
Result of dot product of two vectors is a scalar quantity.
Dot product is commutative : A.B = B.A
Dot product is distributive : A . (B+C) = A.B + A.C also A.A = |A|2

CROSS PRODUCT OF TWO VECTORS
Cross product or vector product of two vectors A and B is defined as
A x B = |A| |B| sinθ nˆ
where nˆ is the unit vector pointing in the direction perpandicular to the plane of both A and B. Result of vector product is also a vector quantity.
Cross product is distributive i.e., A x (B + C) = (A x B) + (A x C) but not commutative and the cross product of two parallel vectors is zero.

In our next post we'll study vector algebra in component form and also lern about vector triple products.

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