brodidit for students
Home / Notes / Class 12 / Physics / Current electricity
Current electricity: Class 12 Physics
Electric current, Ohm's law, resistivity and temperature effects, resistors in series and parallel, EMF and internal resistance, Kirchhoff's rules, Wheatstone bridge, potentiometer, electric power and the colour code.
Key points
Electric current is the rate of flow of charge through a conductor. I = Q/t. Its unit is the ampere (A), where 1 A = 1 C/s. Conventional current is taken in the direction a positive charge would move, from the positive to the negative terminal outside the source. In a metal the real carriers are electrons, which move the opposite way.
Ohm's law: the current through a conductor is directly proportional to the potential difference across it, if the temperature and other physical conditions stay constant. So V = IR. The constant R is the resistance, with unit ohm (Ω), where 1 Ω = 1 V/A. A graph of V against I for an ohmic conductor is a straight line through the origin. A filament lamp, a diode and a thermistor do not obey Ohm's law.
Resistance of a wire depends on:
- Length: R is directly proportional to L.
- Cross-sectional area: R is inversely proportional to A.
- The material, shown by its resistivity ρ, so R = ρL/A.
- Temperature.
Resistivity ρ is the resistance of a wire of unit length and unit cross-section. Its unit is the ohm metre (Ω m). It depends on the material and temperature but not on the size of the wire. Conductivity σ = 1/ρ, with unit Ω⁻¹ m⁻¹ (siemens per metre).
For metals, resistance rises with temperature, because the atoms vibrate more and collide more often with the moving electrons. This is written R = R₀(1 + αΔT), where α is the temperature coefficient of resistance, in K⁻¹ or °C⁻¹. For semiconductors and insulators, resistance falls as temperature rises because more charge carriers become free.
Combination of resistors:
- In series the same current passes through each resistor and the voltages add. R = R₁ + R₂ + R₃. The total is larger than the biggest resistor.
- In parallel the voltage across each is the same and the currents add. 1/R = 1/R₁ + 1/R₂ + 1/R₃. The total is smaller than the smallest resistor.
EMF (electromotive force) of a source is the energy given by the source to each coulomb of charge passing through it. Its unit is the volt. It is measured as the terminal voltage when no current is drawn. A real cell also has internal resistance r. When a current I flows, E = IR + Ir, so the terminal voltage is V = E − Ir. The terminal voltage is less than E when the cell delivers current.
Kirchhoff's rules are used for circuits that cannot be reduced by series and parallel rules:
- First rule (junction rule): the sum of currents entering a junction equals the sum leaving it. This is conservation of charge.
- Second rule (loop rule): around any closed loop the algebraic sum of EMFs equals the algebraic sum of the products IR. This is conservation of energy.
A Wheatstone bridge has four resistors P, Q, R and S in a diamond with a galvanometer across the middle. The bridge is balanced, with zero galvanometer current, when P/Q = R/S. The unknown resistance is then found from the other three. The metre bridge is a practical form that uses a uniform wire of length 100 cm.
A potentiometer is a long uniform wire carrying a steady current. The potential drop along it is proportional to length. It compares EMFs of two cells by the balance lengths: E₁/E₂ = L₁/L₂. It is better than a voltmeter because at balance it draws no current from the cell, so it reads the true EMF.
Electric power is the rate of energy transfer, P = VI = I²R = V²/R, in watts (W). Electrical energy is E = Pt in joules. Domestic bills use the kilowatt hour (kWh), where 1 kWh = 3.6 × 10⁶ J. The heating effect of current is called Joule heating, H = I²Rt.
The colour code gives the value of carbon resistors. The first two bands give digits, the third band gives the power of ten multiplier, and the fourth band gives the tolerance (gold 5%, silver 10%). The digits are black 0, brown 1, red 2, orange 3, yellow 4, green 5, blue 6, violet 7, grey 8, white 9.
Study tip: Learn the definitions of EMF and resistivity word for word, since they are asked often. In numericals, draw the circuit, reduce series and parallel groups step by step, and write the unit with every answer. For Kirchhoff problems, mark current directions and the loop direction before writing equations.
Definitions and flashcards
Say the meaning first, then open the card to check.
Electric current
The rate of flow of electric charge, measured in amperes.
Resistance
The opposition a conductor offers to current, R = V/I, in ohms.
Resistivity
The resistance of a unit length and unit area of a material, in ohm metres.
EMF
The energy supplied by a source per unit charge, in volts.
Internal resistance
The resistance of the material inside a source such as a cell.
Terminal voltage
The potential difference across a source when it delivers current, V = E − Ir.
Ohmic conductor
A conductor whose V against I graph is a straight line through the origin.
Temperature coefficient
The fractional change in resistance per kelvin rise in temperature.
Kilowatt hour
The unit of electrical energy, equal to 3.6 × 10⁶ J.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Current | I = Q / t | Charge Q passing in time t. |
| Ohm's law | V = IR | Valid when temperature is constant. |
| Resistance of a wire | R = ρL / A | ρ is resistivity, L length, A area. |
| Temperature effect | R = R₀(1 + αΔT) | R₀ is the resistance at the reference temperature. |
| Series | R = R₁ + R₂ + R₃ | Equivalent resistance in series. |
| Parallel | 1/R = 1/R₁ + 1/R₂ + 1/R₃ | Equivalent resistance in parallel. |
| Terminal voltage | V = E − Ir | E is EMF, r the internal resistance. |
| Power | P = VI = I²R = V²/R | Rate of energy use in watts. |
| Wheatstone balance | P / Q = R / S | Condition of zero galvanometer current. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
State Ohm's law and write its limitation. (2 marks)
The current through a conductor is directly proportional to the potential difference across its ends, provided temperature and other conditions stay constant. V = IR. It fails for devices such as diodes, thermistors and filament lamps.
Why does the resistance of a metal rise with temperature? (2 marks)
At higher temperature the atoms vibrate with larger amplitude. The free electrons collide with them more often, so their drift is hindered more and the resistance increases.
Define EMF and explain why terminal voltage is less than EMF when a cell supplies current. (3 marks)
EMF is the energy given by the source to each unit charge that passes through it. When a current I flows, some voltage Ir is lost across the internal resistance r of the cell. The terminal voltage is therefore V = E − Ir, which is less than E. When no current is drawn, V equals E.
State Kirchhoff's two rules and the law behind each. (3 marks)
The junction rule says the sum of currents entering a junction equals the sum leaving it, which expresses conservation of charge. The loop rule says the algebraic sum of EMFs in a closed loop equals the algebraic sum of the potential drops IR, which expresses conservation of energy.
Describe the principle of the Wheatstone bridge and write the balance condition. (3 marks)
Four resistors P, Q, R and S form a diamond. A cell is joined across one diagonal and a galvanometer across the other. The resistance R is adjusted until the galvanometer shows no deflection. At this balance the points across the galvanometer are at equal potential and P/Q = R/S. The unknown resistance is found from this ratio.
Why is a potentiometer preferred to a voltmeter for measuring EMF? (5 marks)
A potentiometer is a long uniform wire with a steady current in it, so the potential drop is proportional to the length. The cell to be tested is connected through a galvanometer, and the jockey is moved until the galvanometer shows no deflection. At this balance point no current is drawn from the cell, so there is no loss across its internal resistance and the potentiometer measures the true EMF. A voltmeter always draws a small current and reads the terminal voltage, which is less than the EMF. Two EMFs are compared by E₁/E₂ = L₁/L₂, using their balance lengths.
Numericals with worked steps
A wire of length 2.0 m and cross-sectional area 1.0 × 10⁻⁷ m² has resistivity 1.7 × 10⁻⁸ Ω m. Find its resistance.
- R = ρL/A.
- R = (1.7 × 10⁻⁸ × 2.0) / (1.0 × 10⁻⁷).
- Numerator = 3.4 × 10⁻⁸. Divide by 10⁻⁷ gives 0.34.
Answer: R = 0.34 Ω
Resistors of 6 Ω and 3 Ω are joined in parallel, and a 4 Ω resistor is joined in series with the pair. A 12 V battery of negligible internal resistance is connected across the whole. Find the current from the battery.
- Parallel part: 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2, so R = 2 Ω.
- Total resistance = 2 + 4 = 6 Ω.
- I = V/R = 12 / 6 = 2 A.
Answer: I = 2 A
A cell of EMF 1.5 V and internal resistance 0.5 Ω is connected to a 2.5 Ω resistor. Find the current and the terminal voltage.
- Total resistance = 2.5 + 0.5 = 3.0 Ω.
- I = E/(R + r) = 1.5 / 3.0 = 0.5 A.
- V = E − Ir = 1.5 − 0.5 × 0.5 = 1.5 − 0.25 = 1.25 V.
- Check: IR = 0.5 × 2.5 = 1.25 V.
Answer: I = 0.5 A, terminal voltage = 1.25 V
A 2 kW heater runs for 3 hours at 10 rupees per kWh. Find the energy used and the cost.
- Energy = P × t = 2 kW × 3 h = 6 kWh.
- Cost = 6 × 10 = 60 rupees.
Answer: 6 kWh, cost 60 rupees
Common mistakes
- Adding resistors in parallel directly. Use the reciprocal rule, and check that the result is smaller than the smallest resistor.
- Saying that conventional current is the flow of electrons. Conventional current is opposite to the electron flow.
- Using terminal voltage where EMF is needed. Remember V = E − Ir, and also include r in the total resistance.
- Forgetting to convert units, for example mm² to m² (1 mm² = 10⁻⁶ m²) or cm to m, before using R = ρL/A.
- Writing the wrong sign for an EMF or IR term in Kirchhoff's loop rule. Fix the loop direction first and keep it.