Resistance decides how easily current flows through a wire, and it is one of the highest-yield electricity topics in CDS and OTA Science. In this Cavalier lesson you will learn Ohm’s law, the resistance formula, what resistivity really means, and how to crack series and parallel combination problems quickly — with worked numbers and a previous-year question.
Why Resistance Matters in CDS
Almost every electricity question in CDS Science touches resistance in some way — bulbs, heaters, fuses, household wiring and simple circuits all depend on it. Examiners like this area because a single formula can be twisted into many one-mark questions, from a direct Ohm’s-law calculation to a tricky series-parallel network or a conceptual question on resistivity.
Before we define resistance, recall what electric current is. Current is the rate of flow of charge, measured in amperes (A), where one ampere equals one coulomb of charge passing a point each second. A potential difference (voltage), measured in volts (V), is what pushes that charge through a conductor — rather like the pressure difference that drives water through a pipe.
Resistance, written R, is the property of a conductor that opposes the flow of electric current. Inside a metal, free electrons drift towards the positive terminal but keep colliding with the fixed metal ions; these collisions are the physical origin of resistance and they also convert electrical energy into heat. A good conductor like copper has low resistance; a poor conductor like nichrome has high resistance, which is exactly why nichrome is used in heaters.
The SI unit of resistance is the ohm (Ω). One ohm is the resistance when one volt drives one ampere: 1 Ω = 1 V / 1 A.
Ohm's Law: The Core Relationship
Ohm’s law, given by the German physicist Georg Simon Ohm in 1827, is the backbone of this entire chapter. It states that, at constant temperature, the current I flowing through a conductor is directly proportional to the potential difference V across its ends. In symbols, V ∝ I, and removing the proportionality sign introduces a constant we call resistance.
V = I × R, where V is in volts, I in amperes and R in ohms. Rearranged: I = V / R and R = V / I.
So if you double the voltage across a fixed resistor, the current doubles. If you double the resistance at the same voltage, the current halves. The graph of V (y-axis) against I (x-axis) for an ohmic conductor is a straight line through the origin, and its slope gives the resistance — a steeper line means a larger resistance.
A small but important point: resistance R = V/I is simply the ratio of voltage to current. For an ohmic material this ratio stays constant no matter what voltage you apply, which is the real content of Ohm’s law. For a non-ohmic material the ratio changes, so the law breaks down.
Ohm’s law holds only at constant temperature. A filament bulb is non-ohmic — its resistance rises as it heats up, so its V–I graph curves away from the straight line.
Resistivity and the Factors of Resistance
The resistance of a wire is not a fixed material constant — it depends on the wire’s shape. Four factors control it:
- Length (L): longer wire → more resistance (R ∝ L).
- Area of cross-section (A): thicker wire → less resistance (R ∝ 1/A).
- Material: captured by resistivity ρ.
- Temperature: for metals, resistance rises with temperature.
R = ρ L / A. Here ρ (rho) is the resistivity of the material, with SI unit ohm-metre (Ω·m). Rearranging gives ρ = RA/L.
Resistivity is a property of the material only, independent of the wire’s length or thickness. This is why it is so useful: two copper wires of completely different shapes have different resistances but exactly the same resistivity. Conductors have low resistivity (copper ≈ 1.6 × 10−8 Ω·m, silver slightly lower), insulators have very high resistivity (1012 Ω·m or more), and semiconductors sit in between.
Temperature matters too. For most metals, resistivity and therefore resistance increase as the conductor gets hotter, because the vibrating ions obstruct the electrons more. For semiconductors and electrolytes the opposite happens — resistance falls with rising temperature. Alloys change very little with temperature, which makes them ideal for precision resistors.
If you stretch a wire to double its length, its area halves, so R becomes 4 times the original — a favourite CDS trap.
Conductors, Insulators and Semiconductors
Materials are classified by how freely charge moves through them:
- Conductors: metals like copper, silver, aluminium — many free electrons, low resistivity.
- Insulators: rubber, glass, dry wood, plastic — almost no free electrons, very high resistivity.
- Semiconductors: silicon, germanium — resistivity between the two and falls as temperature rises.
Note that even water and the human body conduct electricity because of dissolved ions, which is why wet hands near a switch are dangerous. Pure distilled water, by contrast, is a poor conductor. Earth itself acts as a huge conductor, and the earth wire in a three-pin plug safely carries leakage current away to prevent shocks.
Silver is the best conductor, but copper and aluminium are used for wiring because they are cheaper. Alloys (nichrome, manganin, constantan) have higher resistivity and resist oxidation, so they are used in heating elements and standard resistors.
Resistors in Series
When resistors are joined end to end so the same current flows through each, they are in series.
Series total: Rs = R1 + R2 + R3 + … The combined resistance is always larger than the biggest single resistor.
Key features of a series circuit:
- Current is the same everywhere — there is only one path for charge to flow.
- Voltage divides across the resistors (more voltage drops across a bigger resistor).
- The total supply voltage equals the sum of the individual voltage drops: V = V1 + V2 + V3.
- If one component fails (open), the whole circuit stops — like old fairy lights where one dead bulb darkens the string.
Series connection is therefore useful when you deliberately want to limit current or share voltage, for example a resistor placed in series with an LED to protect it.
Resistors in Parallel
When resistors are connected across the same two points so each gets the same voltage, they are in parallel.
Parallel total: 1/Rp = 1/R1 + 1/R2 + 1/R3 + … The combined resistance is always smaller than the smallest single resistor.
Key features of a parallel circuit:
- Voltage is the same across every branch.
- Current divides; more current flows through the smaller resistance.
- The total current equals the sum of the branch currents: I = I1 + I2 + I3.
- Each appliance can run independently — this is why home wiring is parallel.
Because adding more parallel branches gives current extra paths, the overall resistance keeps dropping and the total current drawn from the source rises. This is also why connecting too many high-power appliances to one parallel circuit can overload it and blow the fuse.
For just two resistors in parallel, use the shortcut Rp = (R1 × R2) / (R1 + R2) — often remembered as ‘product over sum’. Two equal resistors R in parallel give R/2, and n equal resistors give R/n.
A quick sanity check you can use in the exam: the parallel result must be smaller than the smallest resistor in the group. If your answer comes out larger, you have made an arithmetic slip somewhere and should redo the reciprocals.
Resistance, Power and Heating Effect
When current flows through a resistance, electrical energy turns into heat — the basis of heaters, irons and fuses. This is Joule’s heating.
Power P = VI = I2R = V2/R. Heat produced H = I2R t (joules), where t is time in seconds.
The I2R form explains why high current is dangerous in thin wires, and why electricity is transmitted at high voltage and low current to cut heat losses over long distances. A fuse is a short piece of high-resistance, low-melting-point wire that melts and breaks the circuit when current gets too large, protecting expensive appliances.
The practical unit of electrical energy on your bill is the kilowatt-hour (kWh), also called one ‘unit’. It is the energy used by a 1000-watt device running for one hour, equal to 3.6 × 106 joules. CDS sometimes asks you to combine power and time to find energy consumed, so keep P = VI and energy = P × t ready.
For a fixed voltage supply (like the mains), use P = V2/R: a lower resistance draws more power. That is why a 100 W bulb has lower resistance than a 60 W bulb of the same voltage rating.
Worked Example: Mixed Network
Let us solve a typical mixed-circuit problem step by step.
Two resistors of 6 Ω and 3 Ω are connected in parallel, and this combination is joined in series with a 4 Ω resistor across a 12 V battery. Find the total resistance and the current from the battery.
So the total resistance is 6 Ω and the battery delivers a current of 2 A.
Common Mistakes to Avoid
Adding parallel resistors directly. For parallel you must add the reciprocals first, then invert — do not forget the final flip of 1/Rp.
Mixing up the rules: in series current is shared-equally and voltage divides; in parallel voltage is equal and current divides. Reversing these wrecks the whole answer.
Confusing resistance with resistivity. Resistivity (ρ) depends only on material; resistance (R) also depends on length and area. Stretching a wire changes R but not ρ.
Previous-Year Practice
Try this CDS-style question, then check the worked answer.
Q. Three resistors of 2 Ω, 3 Ω and 6 Ω are connected in parallel. What is the equivalent resistance of the combination?
Answer: 1/Rp = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1, so Rp = 1 Ω. Note how the result is smaller than the smallest resistor (2 Ω), which is the signature of a parallel network.
If equal resistors are in parallel, divide one resistance by the number of resistors. Example: three 6 Ω resistors in parallel give 6/3 = 2 Ω instantly.
Quick Revision
- Ohm’s law: V = IR, at constant temperature.
- Resistance: R = ρL/A; unit ohm (Ω); resistivity unit ohm-metre.
- Resistivity depends on material only, not on shape.
- Series: Rs = R1 + R2 + …; same current, voltage divides.
- Parallel: 1/Rp = 1/R1 + 1/R2 + …; same voltage, current divides.
- Heating: H = I2Rt; power P = VI = I2R = V2/R.
- Home wiring is parallel; a fuse is a high-resistance safety wire.
Frequently asked questions
What is the difference between resistance and resistivity?
Resistance (R) is how much a particular wire opposes current and depends on its length, area and material. Resistivity (ρ) is a property of the material alone, independent of the wire's dimensions, with unit ohm-metre.
Why is household wiring done in parallel and not in series?
In parallel, every appliance gets the same full supply voltage and can be switched on or off independently. In series, all devices would share the voltage and one failed device would cut power to the whole house.
Does the equivalent resistance increase or decrease in parallel?
It decreases. The parallel combination is always smaller than the smallest individual resistor because current gets more paths to flow, lowering overall opposition.
Why is nichrome used in heating elements?
Nichrome has high resistivity and a high melting point, and it does not oxidise easily at high temperatures. The high resistance produces large I²R heat without the wire burning out.
Is Ohm's law valid for all conductors?
No. It holds for ohmic conductors like metals at constant temperature. Devices such as filament bulbs, diodes and semiconductors are non-ohmic, so their V-I graphs are not straight lines.
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