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

Where

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

v

_{d}= 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 = nev

_{d}

**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(T

_{0})[1 + α(T - T

_{0})]

α is called temperature coefficient of resistivity.

6. The work done by the battery force per unit charge is

Є = W/q = F

_{b}*d/q

This Є is called the emf of the battery. Please note that emf is work done/charge.

If nothing is externally connected

F

_{b}= qE or

F

_{b}*d = qEd = qV (because V =Ed)

V = potential difference between the terminals

As Є = W/q = F

_{b}*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.

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