Current Electricity notes

MDCAT Physics

Current electricity deals with the motion of charge, potential difference, resistance, resistivity, circuit laws, electrical sources and power transfer. It also explains practical circuits such as potential dividers and Wheatstone bridges, and the behaviour of conductors when temperature changes.

Electric Current and Steady Current

Electric current is the rate at which electric charge passes through a cross-section of a conductor. If charge Q passes in time t, the current is I = Q/t. The SI unit of current is ampere (A). One ampere is one coulomb of charge passing per second.

A steady current has constant magnitude and direction with time. In metallic conductors, current is due to the drift motion of free electrons. Conventional current is taken to flow from the positive terminal to the negative terminal outside a source, while electrons move in the opposite direction.

A source of emf supplies energy to charges and produces a potential difference in a circuit. The potential difference between two points is the work done or energy transferred per unit charge: V = W/Q.

Current: I = Q/t.
SI unit of current: ampere, A.
Conventional current flows from positive to negative terminal in the external circuit.
Electron flow in a metal is opposite to conventional current.
Steady current has constant magnitude and direction.
Potential difference: V = W/Q, measured in volt (V).
The source of emf converts other forms of energy into electrical energy.
A temperature change can be converted into electrical voltage by a thermocouple or temperature transducer.

Resistance, Ohm's Law and Conductance

Resistance is the opposition offered by a conductor to the flow of electric current. It is defined by R = V/I. Its SI unit is ohm (Ω). A conductor has a resistance of 1 Ω when a potential difference of 1 V produces a current of 1 A through it.

Ohm's law states that the current through a conductor is directly proportional to the potential difference across it, provided physical conditions, especially temperature, remain constant. Thus V = IR. Ohm's law is not applicable to non-ohmic devices or when the temperature changes significantly.

The reciprocal of resistance is conductance. Conductance is represented by G and is given by G = 1/R. Its SI unit is siemens (S), also called mho in older texts.

Ohm's law: V = IR, only when temperature and other physical conditions remain constant.
Resistance: R = V/I.
Conductance: G = 1/R.
Unit of resistance: ohm, Ω.
Unit of conductance: siemens, S.
A 9 V potential difference across 0.2 Ω gives I = 9/0.2 = 45 A.
An ordinary filament bulb is non-ohmic during operation because its temperature changes as current changes.
Resistance depends on length, cross-sectional area, nature of material and temperature.

Resistance and Resistivity

For a uniform wire, resistance is R = ρL/A, where ρ is resistivity, L is length and A is cross-sectional area. Resistivity is a property of the material. It is the resistance of a material specimen having unit length and unit cross-sectional area.

Resistance changes when the dimensions of a wire change, but resistivity does not change merely because the length or area changes. Resistivity changes mainly with the nature of the material and temperature.

If the radius of a wire is doubled, its cross-sectional area becomes four times because A = πr2. If its length is also doubled, the new resistance is R' = ρ(2L)/(4A) = R/2. The resistivity remains unchanged if temperature is unchanged.

Resistance of a wire: R = ρL/A.
Resistivity: ρ = RA/L.
SI unit of resistivity: ohm metre, Ω m.
Resistance increases when length increases.
Resistance decreases when cross-sectional area increases.
Specific resistance is another name for resistivity.
If the length and radius are both doubled, resistance becomes half and resistivity remains unchanged.
Two conductors may have the same resistivity but different resistances because their lengths or cross-sectional areas may differ.
Resistance in series: Rs = R1 + R2 + R3 + ... .
For two equal 6 Ω resistors in parallel, R = 6/2 = 3 Ω.
For parallel resistors, 1/R = 1/R1 + 1/R2 + 1/R3 + ... .
The equivalent resistance of parallel resistors is less than the smallest individual resistance and therefore less than their sum.

Temperature Dependence of Resistance and Resistivity

For many metallic conductors, resistance increases approximately linearly with temperature over a limited range. The relation is Rθ = R0(1 + αθ), where R0 is resistance at 0°C, Rθ is resistance at θ°C and α is the temperature coefficient of resistance at 0°C.

The temperature coefficient is defined as the fractional change in resistance per unit change in temperature. It is measured in K-1. A temperature difference in degrees Celsius has the same numerical size as the difference in kelvin.

Metals generally have a positive temperature coefficient. Semiconductors such as silicon have a negative temperature coefficient of resistivity. Their resistivity decreases when temperature rises because the number of charge carriers increases.

Temperature coefficient: α = change in resistance divided by initial resistance and temperature change.
Unit of α: K-1 or °C-1 for temperature intervals.
For a conductor, Rθ = R0(1 + αθ).
For metals, α is generally positive.
For semiconductors such as silicon, the temperature coefficient of resistivity is negative.
A copper wire has R0 = 10 Ω at 0°C and R10 = 20 Ω at 10°C.
For this wire, α = (20 - 10)/(10 × 10) = 0.1 K-1.
Ohm's law applies only when temperature remains constant.
A filament bulb does not obey Ohm's law during normal operation because its filament temperature changes.

Combination of Cells and Internal Resistance

A practical cell has an emf E and internal resistance r. Internal resistance is the resistance offered by the source of emf itself. It is due to the electrolyte and internal parts of the cell. When current I flows through an external resistance R, part of the emf is lost inside the cell.

The terminal potential difference of a discharging cell is V = E - Ir. Since the external circuit receives current I, V = IR. Therefore, E = I(R + r). For an open circuit, I = 0, so the terminal voltage is approximately equal to the emf.

A cell's internal resistance can be found from E = I(R + r), or r = E/I - R. When a cell is connected directly to an ammeter, the external resistance is the ammeter resistance, if other resistance is neglected.

Internal resistance is the resistance offered by a source of emf.
Emf E is the energy supplied by the source per unit charge.
Terminal voltage while discharging: V = E - Ir.
Complete circuit relation: E = I(R + r).
Short-circuit current: I = E/r, when external resistance is nearly zero.
A 1.5 V cell gives 15 A through an ammeter of resistance 0.04 Ω.
Total resistance is E/I = 1.5/15 = 0.1 Ω.
Internal resistance is 0.1 - 0.04 = 0.06 Ω.
Cells in series add their emfs when connected in the same direction, while their internal resistances also add.
A source's terminal voltage is less than its emf when it supplies current.

Kirchhoff's Laws and Circuit Analysis

Kirchhoff's laws are used for circuits containing several branches and sources. The first law is based on conservation of charge. At any junction, the total current entering equals the total current leaving. In algebraic form, the algebraic sum of currents at a junction is zero.

Kirchhoff's second law is based on conservation of energy. Around any closed loop, the algebraic sum of potential changes is zero. A rise in potential across a source and drops across resistors must be assigned signs according to the chosen direction.

For a junction, currents entering may be taken as positive and currents leaving as negative, or the reverse, provided one convention is used consistently. A relation such as i2 = i1 - i3 - i4 + i5 follows by applying the first law with the assigned current directions.

Kirchhoff's first law is the junction rule.
First law: sum of currents entering a junction equals sum of currents leaving it.
Kirchhoff's second law is the loop rule.
Second law: algebraic sum of potential changes around a closed loop is zero.
The first law follows from conservation of charge.
The second law follows from conservation of energy.
Potential rise through a source and potential drop across a resistor must be given opposite signs.
For a resistor, the potential drop in the direction of current is IR.
A circuit equation should be written after choosing current directions and loop directions.

Wheatstone Bridge and Potential Divider

A Wheatstone bridge consists of four resistances arranged in two branches, a cell across one diagonal and a galvanometer across the other diagonal. It is used to measure an unknown resistance accurately by comparison with known resistances.

When the bridge is balanced, no current flows through the galvanometer. This happens because the two terminals of the galvanometer are at the same potential, so their potential difference is zero and the galvanometer shows zero deflection.

A potential divider is a circuit that provides a continuously variable potential from a fixed supply. For two series resistors R1 and R2, the output voltage across R2 is Vout = V R2/(R1 + R2), provided the output is not significantly loaded.

A balanced Wheatstone bridge measures an unknown resistance.
At balance, the galvanometer shows zero deflection.
At balance, both galvanometer terminals are at the same potential.
For bridge arms P, Q, R and S, balance condition is P/Q = R/S.
The unknown resistance can be calculated by rearranging the balance condition.
A galvanometer detects small current and current direction.
A potential divider gives a selected fraction of the supply voltage.
A variable resistor can be used in a potential divider to obtain continuously varying potential.

Electrical Instruments, Power and Maximum Power Output

An ammeter measures current and is connected in series. It should have very low resistance so that it does not significantly reduce the current. A voltmeter measures potential difference between two points and is connected in parallel. It should have very high resistance so that it draws very little current.

Electrical power is the rate of electrical energy transfer. It is P = VI. Using V = IR, power may also be written as P = I2R or P = V2/R. For a source with emf E and internal resistance r supplying a variable external resistance R, the power delivered to the load is P = I2R.

Maximum power is delivered to the external load when the load resistance equals the internal resistance of the source. Thus maximum power condition is R = r. At this condition, the current is E/(2r), terminal voltage is E/2 and efficiency is 50 percent.

An ammeter measures current and is connected in series.
An ideal ammeter has zero resistance.
A voltmeter measures potential difference and is connected in parallel.
An ideal voltmeter has infinite resistance.
Power: P = VI = I2R = V2/R.
For a source, load power is P = I2R.
Maximum power output occurs when external resistance R equals internal resistance r.
At maximum power, current I = E/(2r).
At maximum power, terminal voltage across the load is E/2.
Efficiency at maximum power transfer is 50 percent.
An AC transformer and a DC generator are not classified as electromechanical measuring instruments in the stated instrument classification.
A galvanometer is an electromechanical instrument because its pointer movement results from magnetic action on a current-carrying coil.

Key terms

Electric current: The rate of flow of electric charge through a conductor, I = Q/t.
Steady current: A current whose magnitude and direction remain constant with time.
Potential difference: The energy transferred or work done per unit charge between two points, V = W/Q.
Electromotive force: The energy supplied by a source per unit charge when driving charge around a complete circuit.
Resistance: The opposition offered by a conductor to the flow of current, R = V/I.
Ohm's law: The law stating that V is directly proportional to I when temperature and physical conditions remain constant.
Conductance: The reciprocal of resistance, G = 1/R, measured in siemens.
Resistivity: The resistance property of a material represented by ρ in R = ρL/A.
Temperature coefficient: The fractional change in resistance or resistivity per unit change in temperature.
Internal resistance: The resistance offered by the source of emf to the flow of current inside the source.
Terminal potential difference: The potential difference across the terminals of a source while it is supplying current.
Kirchhoff's first law: The junction rule stating that the total current entering a junction equals the total current leaving it.
Kirchhoff's second law: The loop rule stating that the algebraic sum of potential changes around a closed loop is zero.
Wheatstone bridge: A four-resistance network used to determine an unknown resistance by balancing two potential divider branches.
Galvanometer: A sensitive instrument used to detect small currents and their direction.
Potential divider: A series resistor circuit used to obtain a fraction or continuously variable value of the supply voltage.
Maximum power transfer: The condition in which a source supplies maximum power to a load when load resistance equals internal resistance.

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