Electricity and Magnetism
2. Ohm's law, resistance and resistivity
Ohm's law: at constant temperature, the current through a conductor is proportional to the potential difference across it: V = IR, so the V–I graph is a straight line through the origin and its slope gives R. Resistance R is measured in ohm (Ω), with 1 Ω = 1 V/A. If 12 V drives 3 A through a wire, R = 4 Ω. Resistance is the opposition to the flow of charge; it comes from collisions of electrons with the atoms of the conductor.
Resistivity: R = ρ l/A, where l is length, A is cross-sectional area and ρ is the resistivity, a property of the material (and temperature) with unit Ω m. Resistance is directly proportional to length and inversely proportional to area. Resistivity does not change when the wire is cut or stretched. Metals such as silver and copper have very low resistivity, so wires are made of copper or aluminium (in increasing order of resistivity: copper, aluminium, tungsten, nichrome, roughly 10⁻⁸ to 10⁻⁶ Ω m). Alloys such as nichrome have high resistivity and do not oxidise easily when hot, so they are used in heating elements. Resistance of a metal rises with temperature.
Worked example. A wire has R = 4 Ω. Doubling the length alone gives 8 Ω. Halving the radius alone makes the area one quarter, so R becomes 16 Ω. But if the same wire is stretched to twice its length, the volume is fixed so the area halves; both effects act and R = 4 × 2 × 2 = 16 Ω.
3. Resistors in series and in parallel
| Property | Series | Parallel |
|---|---|---|
| Current | Same in every resistor | Divides among the branches |
| Potential difference | Divides; V = V₁ + V₂ + … | Same across every branch |
| Equivalent resistance | R = R₁ + R₂ + …, larger than the largest | 1/R = 1/R₁ + 1/R₂ + …, smaller than the smallest |
Worked examples. Series: 2, 3 and 5 Ω give 10 Ω. Parallel: 6 Ω and 3 Ω give 1/R = 1/6 + 1/3 = 1/2, so R = 2 Ω; for two resistors the shortcut is product/sum = 18/9 = 2 Ω, and two equal resistors of 12 Ω in parallel give 6 Ω. Mixed circuit: a 4 Ω resistor in series with the 6 Ω–3 Ω parallel pair has total R = 4 + 2 = 6 Ω; across a 12 V battery the current is 12/6 = 2 A, the voltage across the 4 Ω resistor is 8 V and across the parallel pair 4 V, which drives 4/6 = 2/3 A and 4/3 = 4/3 A through the two branches (sum 2 A).
4. Heating effect, electric power and energy
When current flows through a resistor, electrical energy is converted to heat: Joule's law of heating H = I²Rt (in joules, with I in A, R in Ω, t in s), which with V = IR can be written H = VIt = (V²/R)t. Power is the rate of energy use: P = VI = I²R = V²/R, with unit the watt (W), 1 W = 1 J/s. Energy E = P × t. The commercial unit is the kilowatt-hour (kWh), called one “unit” on a bill: 1 kWh = 3.6 × 10⁶ J.
Worked examples. A 220 V, 100 W bulb has R = V²/P = 220²/100 = 484 Ω and draws I = P/V = 5/11 A, about 0.45 A. A 2 A current through 50 Ω for 10 s produces H = 2² × 50 × 10 = 2000 J. A 2 kW heater used for 3 hours consumes 2 × 3 = 6 kWh = 6 × 3.6 × 10⁶ = 2.16 × 10⁷ J; at a stated tariff of ₹ 5 per unit the cost is ₹ 30. Two resistors of 20 Ω and 30 Ω in parallel across 12 V dissipate 144/20 = 7.2 W and 144/30 = 4.8 W, total 12 W.
- Bulb filament: tungsten, because of its very high melting point; the bulb is filled with an unreactive gas such as nitrogen or argon to prolong its life. Fuse wire: a low-melting-point alloy that melts when too much current flows and breaks the circuit.
- Heating-effect appliances (heater, toaster, iron, geyser) use high-resistivity alloy coils such as nichrome.
5. Household circuits, fuse and earthing
Domestic supply reaches a house by the live wire and the neutral wire, with a potential difference of 220 V between them in India; the supply is alternating current of frequency 50 Hz. The third wire is the earth wire, connected to a metal plate buried deep in the ground. Appliances are connected in parallel across live and neutral. In the main circuit the fuse or MCB is placed in series with the live wire. Domestic circuits are usually of two kinds: a lower-rating circuit (about 5 A) for bulbs and fans and a higher-rating one (about 15 A) for heavy appliances such as a geyser or air-cooler.
- Short circuit: live and neutral touch directly, resistance becomes very small, and the current becomes very large. Overloading: too many appliances on one circuit, so the total current exceeds the safe limit. In both cases the heating effect would overheat the wires, so the fuse melts first.
- Earthing: the metal body of an appliance (iron, toaster, refrigerator) is joined to the earth wire so that any leakage current flows to the ground, the body stays at earth potential and the user does not get a shock.
- AC and DC: a cell gives direct current (constant direction); the mains is AC, which reverses direction every 1/100 s, i.e. 100 times per second at 50 Hz. AC can be sent over long distances with less energy loss than DC.
6. Magnetic field, current and the motor
A magnetic field is the region around a magnet where its force is felt; a compass needle lines up with it. Field lines emerge from the north pole and enter the south pole outside the magnet (and run south to north inside it), forming closed curves; they never cross, and where they are crowded the field is stronger. Oersted found that a current-carrying wire deflects a compass needle: a current produces a magnetic field. For a long straight wire the field lines are concentric circles whose direction is given by the right-hand thumb rule: hold the wire in the right hand with the thumb along the current and the curled fingers show the field direction. The field grows with current and weakens with distance from the wire; a loop of N turns gives N times the field of one turn.
A solenoid is a long coil of many circular turns. The field inside is uniform, along straight parallel lines; outside it resembles a bar magnet, with a north and a south end. Wound on a soft iron core it becomes an electromagnet, strong when the current or turns per unit length increase; soft iron gains and loses magnetism quickly, which suits cranes, relays and bells, while steel keeps it and is used for permanent magnets.
Force on a current-carrying conductor: a conductor in a magnetic field feels a force, greatest when the current is perpendicular to the field and zero when parallel. Fleming's left-hand rule: stretch the thumb, forefinger and middle finger of the left hand mutually perpendicular; if the forefinger points along the field and the middle finger along the current, the thumb gives the direction of the force. For example, with field towards the east and current towards the north, the force on the conductor is downward. The electric motor applies this: a rectangular coil between the poles of a magnet gets forces on its two sides in opposite directions, so it turns; a split-ring commutator reverses the current every half-turn, keeping the turning in the same sense. It converts electrical energy to mechanical energy and is used in fans, mixers and washing machines; more current, stronger magnet or more turns make it stronger.
7. Electromagnetic induction and the generator
Electromagnetic induction: a current is induced in a coil whenever the magnetic field linked with it changes, for example when a magnet moves towards or away from the coil, or when a nearby coil's current is switched on or off. A galvanometer shows the induced current only during the change, and its direction reverses when the motion reverses. The induced current is larger with faster motion, a stronger magnet or more turns.
Fleming's right-hand rule gives the direction of the induced current in a conductor moving in a field: with the right hand's thumb, forefinger and middle finger mutually perpendicular, the thumb shows the motion of the conductor, the forefinger the field, and the middle finger the induced current. The electric generator uses it: a coil rotated in a magnetic field has current induced in it. In an AC generator the coil's ends are joined to two slip rings that rub against brushes, so the output reverses direction every half-rotation; with a split-ring commutator instead, the output is DC. India's mains frequency of 50 Hz means 50 complete cycles per second.
Key takeaways
- I = Q/t (A = C/s); V = W/Q (V = J/C); Ohm's law V = IR; R = ρl/A, where ρ depends only on material and temperature, and stretching a wire to twice its length quadruples R.
- Series: same current, resistances add. Parallel: same voltage, reciprocals add, and the total is less than the smallest branch. Ammeter in series, voltmeter in parallel.
- H = I²Rt; P = VI = I²R = V²/R; 1 kWh = 3.6 × 10⁶ J. In a series connection the larger resistance dissipates more power; in a parallel connection the smaller one does.
- Household circuits are in parallel with a fuse or MCB in series with the live wire; the earth wire carries leakage current away. Indian mains is 220 V, 50 Hz AC.
- A current produces a magnetic field: right-hand thumb rule for a straight wire; a solenoid's inside field is uniform, and with a soft iron core it is an electromagnet.
- Motor: Fleming's left-hand rule (force from current and field), electrical to mechanical energy, split-ring commutator. Generator: Fleming's right-hand rule (induction), mechanical to electrical energy, slip rings for AC.
Practice questions (17)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
A steady current of 5 A flows through a wire for 2 minutes. How much charge passes through the wire in this time?
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Answer: C — 600 C
Q = I × t with t = 2 × 60 = 120 s, so Q = 5 × 120 = 600 C. Forgetting to convert minutes to seconds gives 10 C, and 300 C is the charge for one minute only, which is the tempting slip.Which statement about the connection of measuring instruments in a circuit is correct?
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Answer: A — An ammeter is connected in series and a voltmeter is connected in parallel with the component
An ammeter must carry the whole circuit current, so it is placed in series and has a very low resistance. A voltmeter measures the potential difference across a component, so it is placed in parallel with it and has a very high resistance; reversing the two would change the circuit or give no useful reading.A uniform wire of resistance 4 Ω is stretched uniformly to twice its original length, its volume remaining the same. What is its new resistance?
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Answer: D — 16 Ω
Doubling the length doubles R, but constant volume makes the cross-sectional area half, which doubles R again, so R = 4 × 2 × 2 = 16 Ω. The answer 8 Ω counts only the longer length and ignores the thinning of the wire.A 4 Ω resistor is connected in series with a parallel combination of a 6 Ω and a 3 Ω resistor, across a 12 V battery of negligible internal resistance. What current does the battery supply?
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Answer: B — 2 A
The 6 Ω and 3 Ω in parallel give 6 × 3/(6 + 3) = 2 Ω, so the total resistance is 4 + 2 = 6 Ω and I = 12/6 = 2 A. Adding all three resistors as if in series gives 13 Ω and about 0.92 A, and using 4 Ω alone gives 3 A, which is the tempting error.A uniform metal wire is cut into two equal halves. Which property of the material of the wire is unchanged in each half, and which changes?
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Answer: C — Resistivity is unchanged and resistance is halved
Resistivity belongs to the material and temperature, so it does not change; R = ρl/A is proportional to length, so each half has half the resistance. Saying both halve confuses the material property with the object property, and the doubling option inverts the proportionality.What is the resistance of the filament of an electric bulb rated 220 V, 100 W, when it works at its rated values?
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Answer: A — 484 Ω
From P = V²/R, R = V²/P = 220²/100 = 484 Ω. The value 2.2 Ω comes from V/P-style slips (220/100), and 0.45 Ω is the current in amperes, not a resistance.A 2 kW heater is used for 3 hours. How much electrical energy does it consume, in joules?
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Answer: D — 2.16 × 10⁷ J
E = P × t = 2 kW × 3 h = 6 kWh, and 1 kWh = 3.6 × 10⁶ J, so E = 2.16 × 10⁷ J. The value 7.2 × 10⁶ J is the energy for just one hour, and 2.16 × 10⁴ J slips a factor of 1000 by using power in kW with time in seconds.Consider these statements about household circuits. P: Appliances are connected in parallel so that each gets the full supply voltage. Q: The fuse is connected in series with the live wire. R: The earth wire carries any leakage current from the metal body of an appliance to the ground. S: Fuse wire is made of a material of very high melting point. Which statements are correct?
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Answer: B — P, Q and R only
P, Q and R are correct. S is false: a fuse wire has a low melting point so that it melts and breaks the circuit when excess current heats it; a high-melting-point wire would let the appliance or wiring overheat before it failed.Outside a bar magnet, in which direction do the magnetic field lines run?
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Answer: C — From the north pole to the south pole
By convention field lines emerge from the north pole and enter the south pole outside the magnet, and continue from south to north inside it, forming closed loops. South-to-north is the direction inside the magnet, which is the tempting wrong answer for the outside.Which of the following pairs of a device and the principle it works on is NOT correct?
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Answer: A — Electric motor: electromagnetic induction
A motor works on the force on a current-carrying conductor in a magnetic field; it is the generator that uses induction. The other pairs are right: an electromagnet is the magnetic effect of a current and a fuse melts by Joule heating.A straight conductor carries current towards the north in a uniform magnetic field directed towards the east. Using Fleming's left-hand rule, in which direction is the force on the conductor?
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Answer: D — Downward
Forefinger points east (field) and middle finger north (current); the thumb then points downward, which is the force direction (equal to current × field: north × east is downward). Upward is the result if the field and current are swapped or the right hand is used, which is the common slip.A conductor is moved across a magnetic field and a current is induced in it. Which rule gives the direction of this induced current?
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Answer: B — Fleming's right-hand rule, with the thumb along the motion
Induced current in a moving conductor is found by Fleming's right-hand rule: thumb for motion, forefinger for field, middle finger for current. The left-hand rule gives the force on a current-carrying conductor and is the motor rule; the thumb rule gives the field around a current.Which statement describes the magnetic field inside a long current-carrying solenoid?
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Answer: C — It is uniform and directed along parallel straight lines
Inside a long solenoid the field is uniform and runs along straight parallel lines, and outside it looks like that of a bar magnet. The concentric circles describe a single straight wire, not the interior of a solenoid; the field there adds up and does not cancel.A 20 Ω and a 30 Ω resistor are connected in parallel across a 12 V battery. What is the total power dissipated in the two resistors?
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Answer: A — 12 W
Each resistor has the full 12 V, so the powers are 144/20 = 7.2 W and 144/30 = 4.8 W, totalling 12 W (also V²/R with R = 12 Ω). The value 7.2 W counts only the 20 Ω resistor, and 4.8 W only the 30 Ω one.Two bulbs rated 220 V, 100 W and 220 V, 60 W are connected in series across a 220 V supply. Assuming their resistances stay constant, which statement is correct?
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Answer: D — The 60 W bulb glows brighter because its resistance is greater
Resistance R = V²/P is larger for the 60 W bulb (about 807 Ω against 484 Ω). In series the current is the same, so power I²R is larger in the larger resistance and the 60 W bulb glows brighter. The same current does not mean the same power, and rated power applies only at 220 V across each bulb.In a domestic AC supply in India of frequency 50 Hz, how many times per second does the direction of the current reverse?
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Answer: B — 100 times
Each cycle contains two reversals, one at each half-cycle, so 50 cycles per second means 100 reversals per second, once every 1/100 s. The answer 50 confuses the number of cycles with the number of reversals, and 220 is the voltage.Which of the following lists the materials in increasing order of electrical resistivity at room temperature?
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Answer: C — Copper, aluminium, tungsten, nichrome
Resistivities are roughly copper 1.6 × 10⁻⁸ Ω m, aluminium 2.7 × 10⁻⁸ Ω m, tungsten 5.6 × 10⁻⁸ Ω m and nichrome about 10⁻⁶ Ω m, so the increasing order is copper, aluminium, tungsten, nichrome. The list starting with aluminium swaps the two best conductors, and the one beginning with nichrome is the decreasing order.