1. Average current i(bar) = ∆Q/∆t
The current at time t = I = lim ∆t→0 ( ∆Q/∆t) = dQ/dt
Q is charge, t is time
2. Current density
Average current density j (bar) = Δi/ΔA
i is current and A is area of the conductor
The current density at a point P is
j = lim ∆t→0 (Δi/ΔA) = di/dS
If current i is uniformly distributed over an area S and is perpendicular to it
j = i/S
For a finite area i = ∫j.dS
j = density of current (vector)
dS = area (vector)
3. 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 = nevd
4. Ohm's law
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
5. 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.
6. The work done by the battery force per unit charge is
Є = W/q = Fb*d/q
This Є 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 Є = W/q = Fb*d/q = qV/q = V
Therefore Є = V
7. Thermal energy produced in a register
U = i²Rt
Power developed = P = U/t = i²R = Vi
8. 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)
9. Batteries connected in series
i = (Є 1 + Є 2)/(R + r0)
Where R = external resistance
r0 = r1 + r2 r1, r2 are internal resistances of two batteries
Batteries connected in parallel
Equivalent emf = Є 0 = [Є 1r2 + Є 2r1]/(r1+r2)
where Є 1, Є 2 are emfs of of batteries , and r1, r2 are internal resistances.
equivalent internal restance = r0 = r1r2/(r1 + r2)
So i in the circuit = ε0/(R + r0)
10. Wheatstone Bridge ; R1 and R2 are two resistances connected in series. R3 and R4 are the other two resistance connected series. If the there is no deflection in the galvanometer
R4 = R3R2/R1
as R1/R2 = R3/R4
11. Charging of the capacitor
q = Є (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.
12. Discharging of the capacitor
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.