i = ∆Q/∆t
Q is charge, t is time
j = i/A
i is current and A is area of the conductor
A conductor contains free electrons moving randomly in a lattice of positive ions. Electrons collide with positive ions and their direction changes randomly. In such a random movement, from any area equal numbers of electrons go in opposite directions and due to that no net charge moves and there is no current. But when there is an electric field inside the conductor a force acts on each electron in the direction opposite to the field. The electrons get biased in their random motion in favour of the force. As a result electrons drift slowly in the direction opposite to the field.
If τ be the average time between successive collisions, the distance drifted during this period is
l = ½ a(τ) = ½ (eE/m)( τ) ²
The drift speed is
vd = l/ τ = ½ (eE/m)τ = kE
τ the average time between successive collisions, is constant for a given material at a given temperature.
Relation between current density and drift speed
j = i/A = ne vd
j = σE
E is field and σ is electrical conductivity of the material.
Resistivity of a material ρ = 1/σ
Another form of Ohm's law
V = voltage difference between the ends of a conductor = El (l = length of the conductor)
V = Ri
R = resistance of the conductor = ρ*l/A
1/R is called conductance
Colour codes for resistors
An object of conducting material having a resistance of known value is called a resistor.
Resistors are given colour coding with colour bands that indicate its resistance and tolerance.
Band 1 and band 2 represent digit 1 and digit 2 of resistance.
Band 3 represent the multiplier like 10^n.
Fourth band represents tolerance. No band: 20%, silver 10% and gold 5%.
Temperature dependence of resistivity
As temperature of a resistor increases its resistance increases. The relation can be expressed as
R(T) = R(T0)[1 + α(T - T0)]
α is called temperature coefficient of resistivity.
Thermistors: Measure small changes in temperatures
For these materials resistivity suddenly drops to zero below a certain temperature. For Mercury it is 4.2 K. For the super conducting material if an emf is applied the current will exist for long periods of time even for years without any further application of emf.
Scientists have achieved superconductivity at 125 K so far.
Battery is a device which maintains a potential difference between its two terminals A and B.
In the battery some internal mechanism exerts forces on the charges and drives the positive charges of the battery towards one side and negative charges of towards anther side.
Force on a positive charge q is Fb (a vector quantity). If a charge q is moved from one terminal (say B) to the other terminal say A through an external circuit, the work done by the battery force is Fb*d where d is distance between A and B.
The work done by the battery force per unit charge is
ε = W/q = Fb*d/q
This ε is called the emf of the battery. Please note that emf is work done/charge.
If nothing is externally connected
Fb = qE or
Fb*d = qEd = qV (because V =Ed)
V = potential difference between the terminals
As ε = W/q = Fb*d/q = qV/q = V
Therefore ε = V
Energy transfer in an electric circuit
Thermal energy produced in a resistor
U = i²Rt
Power developed = P = U/t = i²R = Vi
Effect of internal resistance of a battery
Potential difference applied across an external resistor
= emf of the battery - ir (r = internal resistance)
The junction law
The sum of all currents directed towards a point is a circuit is equal to the sum of all the currents directed away from the point.
The loop law
The algebraic sum of all the potential differences along a closed loop in a circuit is zero.
Combination of resistors in series
Equivalent resistance = R1 + R2 +R3+...
Combination of resistors in parallel
1/Equivalent resistance = 1/R1 + 1/R2 +1/R3+...
Division of current in resistors joined in parallel
i1/i2 = R2/R1
i1 = iR2/(R1 + R2)
Batteries connected in series
i = (ε1 + ε2)/(R + r0)
Where R = external resistance
r0 = r1 + r2 r1, r2 are internal resistances of two batteries
Batteries connected in parallel
Equivalent emf =ε0 = ε1r2 + ε2r1/(r1+r2)
where ε1, ε2 are emfs of of batteries , and r1, r2 are internal resistances.
equivalent internal restance = r0 = r1r2/(r1 + r2)
So i in the circurit = ε0/(R + r0)
It is an arrangement of four resistances, and one of them can be measured if the other are known resistances.
R1 and R2 are two resistances connected in series. R3 and R4 are the other two resistance conncted series. If the there is no deflection in the galvanometer
R4 = R3R2/R1
as R1/R2 = R3/R4
Used to measure current in a circuit. A small resistance is connected in parallel to the coil measuring current in an ammeter to reduce the overall resistance of ammeter.
A resistor with a large resistance is connected in series with the coil.
When a volt meter is connected in parallel to the point between which the potential is to be measured, if a large resistance is connected, the equivalent resistance is less than the small resistance.
Charging of the capacitor
q = Є C(1 - e^-t/CR)
q is charge on the capacitor, t is time, Є = emf of the battery, C = capacitance, R is resistance of battery and connecting wires,
CR has units of time and is termed time constant. In one time constant τ (=CR) the charge accumulated becomes 0.63 Є C.
q = Qe^-t/CR
where q is charge remaining on the capacitor
Q is the initial charge
In one time constant 0.63% is discharged.
At about 50 km above the earth’s surface, the air becomes highly conducting and thus there is perfectly conducting surface having potential of 400 kV with respect to earth and current (positive charge) comes down from this surface to earth.
Thunderstorms and lightning bring negative charge to earth.
Water vapour condenses to form small water droplets and tiny ice particles. A parcel of air (cloud) with these droplets and ice particles forms a thunderstorm. A matured thunderstorm is formed with its lower end at a height of 3-4 km above the earth’s surface and the upper end at about 6-7 km above the earth’s surface. Negative charge is at the lower end and positive charge is at the upper end of this thunderstorm. This negative charge creates a potential difference of 20 to 100 MV between these clouds and the earth. This cause dielectric breakdown of air and air becomes conducting.
There are number of thunderstorms every day throughout the earth. They charge the atmospheric batter by supplying negative charge to the earth and positive charge to the upper atmosphere.
It is discharged by this battery.