current electricity

electric current

• it is the rate of flow of electric charge through a cross-section of a conductor

• I = total charge/ by time taken = q/t = dq/dt

• its a scalar quantity

• unit → ampere (A)/ C/s

direction of electric current

• flow of conventional current: direction of the flow of +ve charges

• flow of electronic current: direction of the flow of -ve charges

current density

• it is the amount of charge flowing normally through unit area around that pount

• J = I/ A

• its a vector quantity, whose direction is that of the flow of +ve charges at that point

• unit → A/m²

conduction of electric current

• some outer orbit electrons, called free electrons, can leave their atom and move freely

• they carry the charge in the substance from one place to another

• metals have large no. of free electrons and hence are the best conductors of electricity

• in liquids and gases, conduction takes place by movement of both +ve and -ve charges, while in metals, its only by -ve charges

electric current in the absence and presence of external electric field

• absence:

  • the free electrons move randomly due to thermal velocities

  • the average thermal random velocity is zero

  • hence, theres no net flow of charge in a particular direction

  • thus, no current flows in it

• presence:

  • by attaching positive and negative circular dielectric discs at the end of the cylinder-shaped conductor, an electric field is generated

  • then, the current flows in the direction of the field

ohm’s law

• the current I flowing through a conductor is directly proportional to the potential difference V across its ends, provided that the physical conditions remain constant

V \alpha I

V = IR, (where R = resistance of conductor, constant)

resistance

• it is the ratio of potential difference across the ends to the current flowing through

R = V/I

• unit → ohms (\Omega)/ V/A

• dimensions → [ML²T^-3A^-2]

• it depends on length, shape and nature of material, and is independent of V and I

relation of resistance with length and area (resistivity)

• R is directly proportional to the length of conductor

• R is inversely proportional to the area of cross section

• hence, R \alpha L/A

• R = \rho L/A (where \rho = resistivity/ specific resistance, constant)

• resistivity depends on nature and temperature of conductor

effect of temperature on resistance

• resistance of metallic conductor at temperature t is

• Rt = Ro(1 + \alphat + \betat²)

• where \alpha and \beta are temperature coefficients of resistance

• in most cases, t is small and so \beta = 0

• for metals, \alpha is +ve → resistance increases with rise in temperature

• for insulators, \alpha is -ve → resistance decreases with rise in temperature

ohmic and non-ohmic devices

• ohmic devices are conductors that obey ohm’s law. here, value of R is constant

• non-ohmic devices are insulators, diodes etc. that do not obey ohm’s law. value of R is not constant

• NOTE: R = V/I applies to both ohmic and non-ohmic devices

limitations of ohm’s law

• V may vary non-linearly with I

• the variation of I with V may depend on sign of V applied

• the relation between V and I is not unique (theres more than one value of V for the same I)

(draw the graph for all 3 cases)

drift velocity (vd)

• it is the average velocity with which the electrons move from +ve to -ve end under the presence of an electric field

• it is of the order 10^-4 m/s

relation of vd and E, and its derivation

• vd = eE\tau / m

derive this

relation between vd and I, and its derivation

• I = neAvd

derive this

relation between resistivity, mean relaxation time and density of electrons

\rho = m/ ne²\tau

derive this

deduction of ohm’s law from drift velocity

derive this

mobility

• it is the drift velocity of charge per unit electric field applied

• \mu = eE\tau/m / E = e\tau/m

• unit → m²/Vs

• in practical units → cm²/Vs (order → 10^4)

• it is positive for both positive and negative charges

resistivity of various materials

• it is defined as the resistance of a unit length and unit cross-section of the material of a conductor OR as the resistance of unit cube of a material

• unit → \Omega m

• dimensions → [ML³T^-3A^-2]

conductance

• it is the reciprocal of resistance

• G = 1/R

• unit → mho (\Omega^-1)

relation between J, \sigma and E

J =

also called the microscopic form of ohm’s law

derive this

joule’s heating effect

the amount of heat H produced when current I passes through a conductor of resistance R in time t is given by

H = I²Rt (in J)

joule’s law of heating

let a conductor have resistance R and V is applied to its ends.

current I flowing through it in time t

• total charge flowed → q = It

hence, work done in carrying the charge,

• W = Vq = VIt

• hence, W = H = I²Rt

why are wires of heavy appliances made of nichrome?

• it has high melting point and resistance

• it isnt oxidised easily when heated in air

why are fuse wires made of tin-lead alloy?

• it has low melting point and resistivity

• it has suitable current rating

electrical energy

it is the total work done by emf V in maintaining current I in the circuit for time t

E =W

= I²Rt

hence,

E =VIt = I²Rt = V²t/ R

unit → J

commercial unit → 1 kWh =3.6 × 10^6 J

electrical power

• it is the rate of energy supplied per unit time to maintain the flow of electric current

• P = W/t = VI = I²R = V²/R

unit → W

power is said to be 1W when current 1A flows in against a potential difference 1V

• megawatt (MW) = 10^6 W

• commercial unit, horsepower, 1 HP = 746W

emf of a cell (E)

• it is the work done by the cell in moving a unit +ve charge through the whole circuit

• emf, E =W/q

• unit → J/C or V

NOTE: theres always some fall in emf in the cell due to the current flow through its internal resistance

internal resistance r

• it is the resistance offered by the electrolyte and electrodes of the cell to the current

• it depends on the following factors-

  • directly proportional to the separation between plates off the cell

  • inversely proportional to the area of the plate

  • depends on the nature, concentration and temp. of electrolyte. it increases with increase in concentration

terminal potential difference (V)

• it is the potential difference between the 2 terminals of the cell in a closed circuit

• it is always less than the emf of the cell

relation between V, E and r

• when no current is drawn (cell is in open circuit)

  • I = 0

  • V = E

• when current is drawn

  • current I = E/ R + r

  • hence, equating with ohm’s law, we conclude

  • r = (E/V - 1)R

(derive both with diagrams)

• also, from the definition of potential difference,

  • V = E - Ir

charging of a cell

• during charging, the positive terminal (electrode) is connected to positive terminal of battery, and the negative terminal (electrode) is connected to negative terminal of battery

• in this process, current flows from +ve to -ve electrode of the cell

• hence, the terminal potential becomes V = E + Ir

• the potential drop across r is called lost voltage, and its value = Ir

(draw the diagram)

difference between emf and terminal V

• emf is the maximum potential difference between the 2 terminals of the cell (when its an open circuit), while V is the potential diff. between the 2 terminals of the cell (in a closed circuit)

• emf is independent of resistance of circuit, while V depends on the resistance and the current flowing through its

• the term emf is used for the source of current, while V is measured between any 2 points of the circuit

• emf is a cause, and V is the effect

cells in series

• when the -ve terminal of one cell is connected to +ve terminal of the other cell, its a series combination

• here,

• Eeq = E1 + E2…. + En = nE

• req = r1 + r2…. + rn = nr

• total resistance of the circuit = R + nr

• hence, current in the resistance R, I = nE/ R + nr

NOTE: max current can be drawn if R » nr, or, when the external resistance is very high compared to total internal resistance

cells in parallel

• when the +ve terminals of the cells are connected to the -ve terminals and vice versa, its a parallel combination

• here,

  • Eeq = E

  • 1/rp = 1/r1 + 1/r2 ….. + 1/rm = r/m

  • as R and rp are in series, total resistance in the circuit = R + r/m

  • current in resistance R, I = E/ R + r/m

NOTE:

• max current can be drawn if r/m » R, or, the total internal resistance is very high compared to the external resistance

• Eeq = E because in this combination, only the size of the electrodes increase but not emf

(draw the diagram)

mixed combination of cells

• let there be n cells in series in one row and m rows of cells in parallel

• Eeq of each row = nE

• req of each row = nr

• total r = 1/r = 1/nr + 1/nr …… + up to m times

• hence, r = nr/m

• total resistance = nr/m + R

• current I = nE/ nr/m + R

(draw the diagram)

combination of different cells in series

• let there be n cells of emfs E1, E2… En and internal resistances r1, r2 … rn

• Eeq = E1 + E2 + E3….. + En

• req = r1 + r2 + r3…. + rn

NOTE: if 2 negative terminals are connected (instead of 1 +ve and 1 -ve), then

Eeq = E1 - E2

but req = r1 + r2

combination of cells in parallel

• let there be m cells connected n parallel

• Eeq = E

• 1/req = 1/r1 + 1/r2 + 1/r3 …. 1/rm

kirchoff’s current rule

the algebraic sum of the currents meeting at a point in a circuit is always zero

or,

the sum of currents entering the junction is equal to the sum of currents leaving the junction

• \Sigma I = 0

kirchoff’s voltage law

• the algebraic sum of potential difference around a closed loop is zero

• \Sigma V = 0

sign convention for kirchoff’s laws

• current flowing towards the junction is +ve

• current flowing away from the junction is -ve

• if the direction of the current is opp. from the direction of the loop we choose, V is -ve

• if the direction of the current is same as the direction of the loop we choose, V is +ve

• if the +ve pole of cell is encountered first, V is -ve

• if the -ve pole of cell is encountered first, V is +ve

wheatstone bridge concept

• it is an arrangement of 4 resistances in a parallelogram

• it is used to measure 1 resistance in terms of the other 3

• the bridge is said to be balanced, when the galvanometer gives zero deflection after adjusting the 4 resistances by sending current by the cell (no current flows through the G)

here,

P/Q = R/S

NOTE: when the bridge is balanced, theres no effect if we interchange positions of galvanometer and cell

wheatstone bridge proof

prove this with diagram

wheatstone bridge’s sensitivity

• the sensitivity depends on the values of resistance

• for greater sensitivity, the galvanometer/ battery should be connected across the junctions of 2 highest and 2 lowest resistances

• it is most sensitive when all 4 resistances are equal

advantage of zero deflection in wheatstone bridge

the advantage is that the resistance of G would not affect the balance point. hence, theres no need to determine current in resistances and internal resistance of the G