In case of flow of charge through a cross-section, current density is defined as a vector having Magnitude equal to current per unit area surrounding that point. Remember area is held normal to the direction of charge flow (or current passes) through that point. Current density at point P i.s given by.

`J = (di)/(dA)hatn`
Current density `J` is a vector quantity having Sl unit `A//m^2` and dimension `[L^(-2) A].`
case of uniform flow of charge through a cross-section normal to it as `i = nqvA so, J= i/A hatn =(nqv) hatn`
Current density related with electric field is `J = sigmaE = E/rho` where `sigma =`conductivity and `rho =` resistivity or Specific resistance of substance. The direction of current density `J` is same as that of electric field `E.`

drift velocity is the average uniform velocity acquired by electrons inside a metal by the
application of an electric which is responsible for current through it. Drift Velocity is very small ,it is of the order of `(~~10^(-4) m//s)` as compared to thermal speed `(~~ 10^5 m// s)` of electrons at room temperature. The current related with drift velocity is `i = n eAv_d`

We can also write `v_d = (i)/(n e A) = (J)/(n e)= (sigmaE)/(n e) = (E)/(rhon e) = V/(rhol n e)`
where, `n=` number of electrons per unit volume of the conductor
`A=` area of cross-section
`V =` potential difference across the conductor
`E = `electric field inside the conductor
`i =`current, `J =` current density, `rho =` specific resistance

`sigma =` conductivity (`sigma= 1/rho`)

The electric potential at a point in an electric field is defined as the amount of work done in bringing a unit positive charge from infinity to that point.

.. Note Variable DC potential is represented by the following symbol.

`text(Potential Difference)`
The potential difference between two points in an electric field is defined as the amount of work done in moving a unit postie charge from one point the other point.

According to Ohm's law, physical conditions (temperature, mechanical strain, etc.) remaining unchanged, the current flowing through a conductor is directly proportional to the potential difference across its ends.

Thus, `I propto V` or `V= IR`

where, R is a constant known as the electrical resistance of given conductor.

Electrical resistance is defined as the ratio in the potential difference `(V)` across the ends of the conductor to the current `(I)` flowing through it,

i.e. `R = V/I`
The SI unit of electrical resistance is omega (ohm) and its dimension is `[ML^2 T^(-3) A^(-2)].`

`text(Effect of Temperature on the Resistance)`

(i) On increasing the temperature of the metal, its resistance increases.
(ii) On increasing the temperature of semiconductor its resistance decreases.
(iii) On increasing the temperature of alloy, their resistance increase but it is small compared to the pure metals.
(iv) On increasing the temperature of electrolytes, their resistance decreases.

Ratio of electric field and the current density at a point in the conductor is called the specific resistance or resistivity. It is a constant for the material.

`rho= E/J` or `rho = (RA)/l`
Its unit is ohm-metre.
Some factors affecting specific resistance are given below
(i) Resistivity increases with increases of temperature.
(ii) Resistivity increases on mixing of impurity and increase of mechanical stress.
(iii) The resistance of super conductor is zero.

`text(Conductivity)` Inverse of specific resistance is called conductivity `(sigma)`. Unit of conductivity is mho/metre.
`text(Conductance)` Increase of resistance is called conductance. Unit of conductance is `ohm^(-1)` or siemen or mho.

`text(Different electrical conducting materials of specific uses)`
`(i)` The filament of bulb is made of tungsten. The resistivity and melting point of tungsten both are high.
`(ii)` In heating equipments (heater, gyser and electric press), the coils are made of nichrome because the resistivity and melting of nichrome is very high.
`(iii)` In resistance box, the standard resistances are made of manganin or onstantan. Their resistivity do not depend upon temperature, therefore the value of standard resistance does not vary.
`(iv)` Fuse wire is made of mixed conductor (63%, tin and 37% lead). Th,; value of melting point is low but resistivity is very high. The fuse wire is placed in the series to protect the main line.

`(i) text(Series combination)` In this combination, resistances are joined end to end.

In series combination, equivalent resistance is equal to sum of individual resistance.
`R =R_1 + R_2 + R_3`
• In series combination, current through each resistance is same.
`i = i_1 = i_2 = i_3`
• In series combination potential difference across each resistance is different.
`V= V_1 + V_2 + V_3`
Such that `V_1 : V_2 : V_3 = R_1 : R_2 : R_3`

`(ii) text(Parallel combination)` In this combination, first ends of all the resistances is connected to one point and second ends of all the resistances is connected to other point.
• In parallel combination, equivalent resistance is given by`1/R = (1)/(R_1) + (1)/(R_2) + (1)/(R_3)`
• In parallel combination, potential difference across each resistance is same. `V= V_1 =V_2 =V_3`
• In parallel combination current in each resistance is different
`i = i_1 + i_2 + i_3` such that `i_1 : i_2 : i_3 = (1)/(R_1) : (1)/(R_2) : (1)/(R_3)`

(i) `text(In series)`

In this combination of cells negative end of each cell is connected to the positive end of the next cell.

Let n cells each of internal resistance `r` and emf `E` are connected in series.

So, total emf `= nE ` and total internal resistance `= nr.`

Current taken from the combination

`i = (nE)/(nr +R)`

• If `nr < R,` i.e. total internal resistance of cell is less than the external resistance, then the current taken from the combination is n times the current taken from one cell.

• If `nr > > R.` i.e. total internal resistance of cells is greater than the external resistance. then the current taken from the combination is same as that from one cell. So, there is no benefit, joining the cells of high internal resistance in series order.

(ii) `text(In parallel ) ` In this combination positive terminals of all the cells arc connected to one point and negative terminals to the other point.

• Let `n` cells each of internal resistance rand of emf `E` are connected in parallel. Then. emf of combination `= E` and total internal resistance `= r/n`

Current taken from the combination

`i = (nE)/(r + nR)`

• If `r/n >> R` i.e. total internal resistance of cell is greater than external resistance, then current taken from the combination is n times that from one cell.

• If `r/n >> R` i.e. total internal resistance of cell is less than external resistance, then current taken from the combination is equal to that from one cell. So, there is no benefit of connecting cells of low internal resistance in parallel.

`(iii) text(Mixed grouping)` In this combination, a certain number of cells are connected in various series and all such series are then connected in parallel. Number of cells `= mn`

emf of combination `= nE`

Total internal resistance `= (nr)/m`

Current taken in external resistance,

`i = (mnE)/( nr + mR)`

For maximum current in the circuit mR = nr.

If two cells oppose each other, then current taken from them `I = (E_1 -E_2)/(r_1 +r_2 +R)`

Similarly, if cells support each other, then `I = (E_1 +E_2)/(r_1 +r_2 +R)`

Charging current of a cell is given by

`i = text(emf of charger - emf of cell)/text(total resistance of the circuit)`

In a battery of N cells each of emf E, if n cells arc wrongly connected, then net emf,

`E' =NE - 2nE`