Stretch a Wire to Twice Its Length and Its Resistance Quadruples
See what resistance depends on, calculate it from resistivity and compare conductors with alloys, derive the equivalent resistance for resistors in series and in parallel, and find out why every house is wired in parallel.
Why does a thin long wire resist current more than a short thick one?
Resistance is not a mysterious property that some materials happen to have. It is the result of electrons colliding with the atoms of the conductor as they drift, and everything that affects how often those collisions happen affects the resistance.
That single picture predicts all four dependences at once:
- A longer wire means more atoms to get past, so more collisions — resistance increases with length
- A thicker wire gives the electrons more room, so fewer collisions per electron — resistance decreases as the area increases
- A different material has its atoms packed differently and holds its electrons differently — resistance depends on the material
- A hotter conductor has atoms vibrating more violently, so they are harder to get past — the resistance of a metal increases with temperature
And one of those has a consequence students find surprising. Stretch a wire to twice its length and its resistance becomes four times as large, not twice — because the same material has to fill twice the length, so the cross-section is halved as well. Length doubling and area halving each double the resistance, and the two effects multiply.
Once a single resistor is understood, the chapter turns to combinations. Connecting resistors in series makes the total larger; connecting them in parallel makes it smaller than any of them — and the second fact is why your house is wired the way it is.
This page covers the second part of the CBSE Class 10 Science chapter on electricity: the factors affecting resistance, resistivity, and equivalent resistance in series and in parallel.
That single picture predicts all four dependences at once:
- A longer wire means more atoms to get past, so more collisions — resistance increases with length
- A thicker wire gives the electrons more room, so fewer collisions per electron — resistance decreases as the area increases
- A different material has its atoms packed differently and holds its electrons differently — resistance depends on the material
- A hotter conductor has atoms vibrating more violently, so they are harder to get past — the resistance of a metal increases with temperature
And one of those has a consequence students find surprising. Stretch a wire to twice its length and its resistance becomes four times as large, not twice — because the same material has to fill twice the length, so the cross-section is halved as well. Length doubling and area halving each double the resistance, and the two effects multiply.
Once a single resistor is understood, the chapter turns to combinations. Connecting resistors in series makes the total larger; connecting them in parallel makes it smaller than any of them — and the second fact is why your house is wired the way it is.
This page covers the second part of the CBSE Class 10 Science chapter on electricity: the factors affecting resistance, resistivity, and equivalent resistance in series and in parallel.
What exactly does the resistance of a conductor depend on?
On its length, its area of cross-section, the material it is made of, and its temperature.
Length. Resistance is directly proportional to length:
So doubling the length doubles the resistance, and a wire cut in half has half the resistance of the original.
Area of cross-section. Resistance is inversely proportional to the area:
So a wire of twice the cross-sectional area has half the resistance. That is why the thick cable to a heavy appliance is thick — a thin one would resist too much and waste energy as heat.
Material. Copper and nichrome wires of identical length and thickness have very different resistances, because the property belongs to the substance. That property is called resistivity, and it is the subject of the next section.
Temperature. For a metal the resistance increases as the temperature rises, because the atoms vibrate more and obstruct the drifting electrons more often.
Worked example — the stretched wire. A wire has a resistance of ohms. It is stretched uniformly until its length is doubled. Find its new resistance.
The volume of material is unchanged, so if the length doubles the area must halve:
Worked example 2 — cutting instead of stretching. The same ohm wire is cut into two equal halves. Find the resistance of one half, and of the two halves connected side by side.
Each half has half the length and the same area, so each is ohms. Placing them side by side doubles the area and halves the length again:
Notice how different cutting and stretching are. Stretching quadruples the resistance; cutting and doubling up reduces it to a quarter. In both cases the amount of material is the same — only its shape has changed, and the shape is what resistance depends on.
The temperature dependence is what makes Ohm's law conditional. A filament lamp's resistance when cold is far lower than when it is glowing, which is why the current surges briefly at switch-on and why a bulb usually fails at the moment it is switched on rather than in the middle of use. The cold filament briefly allows a much larger current than its working value.
Length. Resistance is directly proportional to length:
So doubling the length doubles the resistance, and a wire cut in half has half the resistance of the original.
Area of cross-section. Resistance is inversely proportional to the area:
So a wire of twice the cross-sectional area has half the resistance. That is why the thick cable to a heavy appliance is thick — a thin one would resist too much and waste energy as heat.
Material. Copper and nichrome wires of identical length and thickness have very different resistances, because the property belongs to the substance. That property is called resistivity, and it is the subject of the next section.
Temperature. For a metal the resistance increases as the temperature rises, because the atoms vibrate more and obstruct the drifting electrons more often.
Worked example — the stretched wire. A wire has a resistance of ohms. It is stretched uniformly until its length is doubled. Find its new resistance.
The volume of material is unchanged, so if the length doubles the area must halve:
Worked example 2 — cutting instead of stretching. The same ohm wire is cut into two equal halves. Find the resistance of one half, and of the two halves connected side by side.
Each half has half the length and the same area, so each is ohms. Placing them side by side doubles the area and halves the length again:
Notice how different cutting and stretching are. Stretching quadruples the resistance; cutting and doubling up reduces it to a quarter. In both cases the amount of material is the same — only its shape has changed, and the shape is what resistance depends on.
The temperature dependence is what makes Ohm's law conditional. A filament lamp's resistance when cold is far lower than when it is glowing, which is why the current surges briefly at switch-on and why a bulb usually fails at the moment it is switched on rather than in the middle of use. The cold filament briefly allows a much larger current than its working value.
Formula
How do you calculate resistance from resistivity?
Multiply the resistivity by the length and divide by the area of cross-section.
where is the resistivity of the material, the length in metres and the area in square metres. Rearranging shows the unit of resistivity:
Resistivity is a property of the material alone, independent of the shape of the piece, which is exactly why it is useful: it lets you compare substances rather than particular wires.
Worked example 1. A copper wire of length m has a cross-sectional area of m. Taking the resistivity of copper as ohm metre, find its resistance.
A very small resistance for a long wire, which is why copper is used for connecting wires.
Worked example 2 — finding the resistivity. A wire m long with an area of m has a resistance of ohms. Find its resistivity.
That is far higher than copper's, so the wire is not a good conductor — it is the kind of value an alloy has.
How the three classes of material compare.
- Metals and alloys — resistivity between about and ohm metre. Excellent conductors
- Insulators such as rubber and glass — resistivity of the order of to ohm metre
- The gap between them is enormous — roughly twenty powers of ten, which is why the same substance is never used for both jobs
Silver and copper have the lowest resistivities among the metals, which is why copper and aluminium are used for electrical transmission.
Why alloys are used for heating elements rather than pure metals. Nichrome, an alloy, has a resistivity much higher than copper's, and two properties make it the right choice:
- Its high resistivity means a short element can produce a large resistance, so a lot of heat is generated in a small space
- It does not oxidise readily even when glowing hot, so the element survives repeated heating
That is why the element of an electric heater is nichrome and its connecting cord is copper — the same current passes through both, and the heat is deliberately produced in one and deliberately avoided in the other. One circuit, two materials, chosen for opposite reasons.
where is the resistivity of the material, the length in metres and the area in square metres. Rearranging shows the unit of resistivity:
Resistivity is a property of the material alone, independent of the shape of the piece, which is exactly why it is useful: it lets you compare substances rather than particular wires.
Worked example 1. A copper wire of length m has a cross-sectional area of m. Taking the resistivity of copper as ohm metre, find its resistance.
A very small resistance for a long wire, which is why copper is used for connecting wires.
Worked example 2 — finding the resistivity. A wire m long with an area of m has a resistance of ohms. Find its resistivity.
That is far higher than copper's, so the wire is not a good conductor — it is the kind of value an alloy has.
How the three classes of material compare.
- Metals and alloys — resistivity between about and ohm metre. Excellent conductors
- Insulators such as rubber and glass — resistivity of the order of to ohm metre
- The gap between them is enormous — roughly twenty powers of ten, which is why the same substance is never used for both jobs
Silver and copper have the lowest resistivities among the metals, which is why copper and aluminium are used for electrical transmission.
Why alloys are used for heating elements rather than pure metals. Nichrome, an alloy, has a resistivity much higher than copper's, and two properties make it the right choice:
- Its high resistivity means a short element can produce a large resistance, so a lot of heat is generated in a small space
- It does not oxidise readily even when glowing hot, so the element survives repeated heating
That is why the element of an electric heater is nichrome and its connecting cord is copper — the same current passes through both, and the heat is deliberately produced in one and deliberately avoided in the other. One circuit, two materials, chosen for opposite reasons.
How do you find the equivalent resistance of resistors in series?
Add them. The equivalent resistance of a series combination is the sum of the individual resistances.
Where that comes from. In a series circuit there is only one path, so:
- The current is the same through every resistor
- The potential differences add up to the total across the combination
So , and using Ohm's law on each term with the same current :
The current cancels, which is why the result is so simple — and the derivation is worth writing out, because the marks for it are separate from the marks for using it.
Worked example. Resistors of , and ohms are connected in series across a V battery. Find the equivalent resistance, the current, and the potential difference across each resistor.
The same A passes through each, so
Check: V, the full battery voltage. That addition is the compulsory check on every series answer.
Notice which resistor takes the largest share. The ohm resistor drops three times as much voltage as the ohm one, because the same current is flowing through all three. In series, the largest resistance gets the largest potential difference — which is worth remembering as a sanity check.
Two properties of the result.
- ** is always larger than the largest individual resistance
- Adding another resistor in series always reduces the current, since the total resistance rises
The practical drawback of series wiring, and it is the reason it is not used in houses. There is only one path, so if any one component fails, the circuit breaks and everything stops. A string of decorative lamps wired in series goes completely dark when one lamp fails, and finding the faulty one means testing them one at a time. Every appliance also shares the supply voltage**, so none of them receives the full V they were designed for.
Where that comes from. In a series circuit there is only one path, so:
- The current is the same through every resistor
- The potential differences add up to the total across the combination
So , and using Ohm's law on each term with the same current :
The current cancels, which is why the result is so simple — and the derivation is worth writing out, because the marks for it are separate from the marks for using it.
Worked example. Resistors of , and ohms are connected in series across a V battery. Find the equivalent resistance, the current, and the potential difference across each resistor.
The same A passes through each, so
Check: V, the full battery voltage. That addition is the compulsory check on every series answer.
Notice which resistor takes the largest share. The ohm resistor drops three times as much voltage as the ohm one, because the same current is flowing through all three. In series, the largest resistance gets the largest potential difference — which is worth remembering as a sanity check.
Two properties of the result.
- ** is always larger than the largest individual resistance
- Adding another resistor in series always reduces the current, since the total resistance rises
The practical drawback of series wiring, and it is the reason it is not used in houses. There is only one path, so if any one component fails, the circuit breaks and everything stops. A string of decorative lamps wired in series goes completely dark when one lamp fails, and finding the faulty one means testing them one at a time. Every appliance also shares the supply voltage**, so none of them receives the full V they were designed for.
How do you handle resistors in parallel, and why is a house wired that way?
Add the reciprocals. The reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.
Where that comes from. In a parallel circuit:
- The potential difference is the same across every resistor, since each is connected between the same two points
- The currents in the branches add up to the total current
So , and using Ohm's law on each term with the same :
This time the voltage cancels, which is the mirror image of the series derivation.
Worked example 1. A ohm and a ohm resistor are connected in parallel across a V battery. Find the equivalent resistance, the total current, and the current in each branch.
Each branch has the full V across it:
Check: A, the total current. That addition is the compulsory check on every parallel answer.
Worked example 2 — three resistors. Find the equivalent resistance of , and ohms in parallel.
**Notice that ohms is smaller than the smallest of the three. That is always true:
- is always less than the smallest individual resistance
- Adding another resistor in parallel always reduces the total resistance and therefore increases the total current
- The smallest resistance carries the largest current, since all branches share the same voltage
Now the reason a house is wired in parallel, which is the part of the chapter examined most often. Four reasons, each following from one of the properties above:
- Every appliance receives the full supply voltage** of V, which is what each was designed for
- Each appliance can be switched on and off independently, because its branch can be broken without breaking the others
- If one appliance fails, the rest keep working — there is no single path to interrupt
- Each appliance draws only the current it needs, according to its own resistance
And the trade-off to state honestly. Because parallel connection lowers the total resistance, connecting more and more appliances draws more and more current from the supply. That is overloading, and it is why a house needs fuses — the subject of Part 3.
Where that comes from. In a parallel circuit:
- The potential difference is the same across every resistor, since each is connected between the same two points
- The currents in the branches add up to the total current
So , and using Ohm's law on each term with the same :
This time the voltage cancels, which is the mirror image of the series derivation.
Worked example 1. A ohm and a ohm resistor are connected in parallel across a V battery. Find the equivalent resistance, the total current, and the current in each branch.
Each branch has the full V across it:
Check: A, the total current. That addition is the compulsory check on every parallel answer.
Worked example 2 — three resistors. Find the equivalent resistance of , and ohms in parallel.
**Notice that ohms is smaller than the smallest of the three. That is always true:
- is always less than the smallest individual resistance
- Adding another resistor in parallel always reduces the total resistance and therefore increases the total current
- The smallest resistance carries the largest current, since all branches share the same voltage
Now the reason a house is wired in parallel, which is the part of the chapter examined most often. Four reasons, each following from one of the properties above:
- Every appliance receives the full supply voltage** of V, which is what each was designed for
- Each appliance can be switched on and off independently, because its branch can be broken without breaking the others
- If one appliance fails, the rest keep working — there is no single path to interrupt
- Each appliance draws only the current it needs, according to its own resistance
And the trade-off to state honestly. Because parallel connection lowers the total resistance, connecting more and more appliances draws more and more current from the supply. That is overloading, and it is why a house needs fuses — the subject of Part 3.
Exam tip
What layout keeps a combination-of-resistors answer safe?
Say whether the connection is series or parallel, write the right formula, and finish with the addition check. The check is what turns a plausible answer into a certain one.
- Name the connection first. Series adds resistances; parallel adds reciprocals
- Take the reciprocal at the very end in a parallel calculation. Forgetting it leaves you with and calling it — the most frequent error in this chapter
- Check the magnitude: is larger than the largest resistor, and is smaller than the smallest. An answer outside those bounds is wrong
- In series, check that the potential differences add to the supply voltage
- In parallel, check that the branch currents add to the total current
- Remember what is common: the current is the same in series, the voltage is the same in parallel
- Convert areas to square metres before using , and give resistivity in ohm metre
- Give all four reasons when asked why domestic circuits are parallel
The misconception to name. Connecting more appliances does not make the supply work harder against a bigger resistance — it makes the total resistance smaller, so the current drawn goes up. More appliances in parallel means more current, not less, and that is exactly why overloading is a hazard and why the fuse exists.
- Name the connection first. Series adds resistances; parallel adds reciprocals
- Take the reciprocal at the very end in a parallel calculation. Forgetting it leaves you with and calling it — the most frequent error in this chapter
- Check the magnitude: is larger than the largest resistor, and is smaller than the smallest. An answer outside those bounds is wrong
- In series, check that the potential differences add to the supply voltage
- In parallel, check that the branch currents add to the total current
- Remember what is common: the current is the same in series, the voltage is the same in parallel
- Convert areas to square metres before using , and give resistivity in ohm metre
- Give all four reasons when asked why domestic circuits are parallel
The misconception to name. Connecting more appliances does not make the supply work harder against a bigger resistance — it makes the total resistance smaller, so the current drawn goes up. More appliances in parallel means more current, not less, and that is exactly why overloading is a hazard and why the fuse exists.
Did you know
Why is the heater element nichrome and its cord copper?
Open up any electric heater and you find two very different wires carrying exactly the same current. The flexible cord from the plug is copper; the glowing coil inside is nichrome. Both were chosen deliberately, and for opposite reasons.
The cord must not get hot. Heat is produced wherever there is resistance, so a connecting wire is made of a material with the lowest resistivity available at a reasonable cost — copper, at about ohm metre — and made thick enough that its resistance is a small fraction of an ohm. The energy then passes through it almost untouched.
The element must get hot. Nichrome's resistivity is far higher, so a coil of modest length has a large resistance, and that is where the energy is deliberately converted into heat. Two properties make it the right alloy:
- High resistivity, so a compact coil gives a large resistance
- It does not oxidise readily when red hot, so it survives being heated and cooled thousands of times
A pure metal of similar resistivity would burn away. That is the real reason alloys rather than elements are used for heating, and it is the sentence an examiner is looking for.
The same reasoning appears in a fuse, with the choice reversed again. A fuse wire is made of a material with a low melting point, so that it melts and breaks the circuit before anything else in the house is damaged. Three wires in one appliance, three different requirements — carry the current without heating, heat up without burning away, and melt before anything else does.
And the coil's shape is chosen too. Winding the element into a long coil packs a large length into a small space, which gives a large resistance and a large heating effect in a compact appliance. Length is one of the four factors from the first section, used on purpose.
One observation that ties it together. Feel the cord of a working heater and it is barely warm; the element is glowing. Same current, same time, different resistance — and the heat produced depends on the resistance, which is exactly what Joule's law in Part 3 will state as a formula.
The cord must not get hot. Heat is produced wherever there is resistance, so a connecting wire is made of a material with the lowest resistivity available at a reasonable cost — copper, at about ohm metre — and made thick enough that its resistance is a small fraction of an ohm. The energy then passes through it almost untouched.
The element must get hot. Nichrome's resistivity is far higher, so a coil of modest length has a large resistance, and that is where the energy is deliberately converted into heat. Two properties make it the right alloy:
- High resistivity, so a compact coil gives a large resistance
- It does not oxidise readily when red hot, so it survives being heated and cooled thousands of times
A pure metal of similar resistivity would burn away. That is the real reason alloys rather than elements are used for heating, and it is the sentence an examiner is looking for.
The same reasoning appears in a fuse, with the choice reversed again. A fuse wire is made of a material with a low melting point, so that it melts and breaks the circuit before anything else in the house is damaged. Three wires in one appliance, three different requirements — carry the current without heating, heat up without burning away, and melt before anything else does.
And the coil's shape is chosen too. Winding the element into a long coil packs a large length into a small space, which gives a large resistance and a large heating effect in a compact appliance. Length is one of the four factors from the first section, used on purpose.
One observation that ties it together. Feel the cord of a working heater and it is barely warm; the element is glowing. Same current, same time, different resistance — and the heat produced depends on the resistance, which is exactly what Joule's law in Part 3 will state as a formula.
Exam relevance
How are series and parallel combinations examined in JEE and NEET?
This is foundation work for Class 12 Current Electricity, examined heavily in JEE Main, JEE Advanced and NEET.
Where the combinations lead. Class 12 extends series and parallel to complicated networks solved with Kirchhoff's laws, and to the Wheatstone bridge and the metre bridge, where a balanced condition is found from four resistances. Every one of those problems begins by reducing parts of the network with the two rules you learn here, and JEE Advanced sets networks where recognising a parallel pair is the whole difficulty.
Where resistivity leads. Class 12 relates it to conductivity, its reciprocal, and then to the microscopic picture: depends on the number density of electrons and on the average time between collisions. The collision explanation you use here for the temperature dependence is the qualitative form of that theory, and the temperature coefficient of resistance is a numerical topic there.
Where the stretched-wire result leads. It is a standard JEE Main one-liner: a wire stretched to times its length has times the resistance, because the volume is constant. **The Class 10 case of giving four times is the same argument, and drawing a wire out to a different area without changing its volume appears repeatedly.
Where the domestic-circuit reasoning leads. Class 12 treats internal resistance of a cell, which is itself a series resistance, and the conditions for maximum power transfer. The habit of asking what is common — current or voltage — is what makes those derivations manageable.
Question types to expect.** At this level: equivalent resistance, branch currents, resistivity numericals, and the reasons for parallel wiring. In competitive papers: network reduction, bridge balance conditions, stretched-wire problems, and assertion-reason items on why is smaller than the smallest resistor.
The single trap that costs marks. Forgetting to invert at the end of a parallel calculation. ** means ohms, not ohm** — and the magnitude check catches it, since must be below the smallest resistor.
A second trap. Assuming a stretched wire has twice the resistance. It has four times, because the area halves as the length doubles. In JEE the exponent is exactly what is being tested.
Board versus competitive emphasis. The CBSE paper marks the derivation, the formula, the substitution and the addition check; a competitive paper marks the reduced network value. **The transferable habit is asking what is common here before writing anything** — the current in series, the voltage in parallel — because that single question generates both derivations.
Where the combinations lead. Class 12 extends series and parallel to complicated networks solved with Kirchhoff's laws, and to the Wheatstone bridge and the metre bridge, where a balanced condition is found from four resistances. Every one of those problems begins by reducing parts of the network with the two rules you learn here, and JEE Advanced sets networks where recognising a parallel pair is the whole difficulty.
Where resistivity leads. Class 12 relates it to conductivity, its reciprocal, and then to the microscopic picture: depends on the number density of electrons and on the average time between collisions. The collision explanation you use here for the temperature dependence is the qualitative form of that theory, and the temperature coefficient of resistance is a numerical topic there.
Where the stretched-wire result leads. It is a standard JEE Main one-liner: a wire stretched to times its length has times the resistance, because the volume is constant. **The Class 10 case of giving four times is the same argument, and drawing a wire out to a different area without changing its volume appears repeatedly.
Where the domestic-circuit reasoning leads. Class 12 treats internal resistance of a cell, which is itself a series resistance, and the conditions for maximum power transfer. The habit of asking what is common — current or voltage — is what makes those derivations manageable.
Question types to expect.** At this level: equivalent resistance, branch currents, resistivity numericals, and the reasons for parallel wiring. In competitive papers: network reduction, bridge balance conditions, stretched-wire problems, and assertion-reason items on why is smaller than the smallest resistor.
The single trap that costs marks. Forgetting to invert at the end of a parallel calculation. ** means ohms, not ohm** — and the magnitude check catches it, since must be below the smallest resistor.
A second trap. Assuming a stretched wire has twice the resistance. It has four times, because the area halves as the length doubles. In JEE the exponent is exactly what is being tested.
Board versus competitive emphasis. The CBSE paper marks the derivation, the formula, the substitution and the addition check; a competitive paper marks the reduced network value. **The transferable habit is asking what is common here before writing anything** — the current in series, the voltage in parallel — because that single question generates both derivations.
Key takeaways
What should you know before the heating effect of current?
Four factors, one formula and two combination rules.
- Resistance increases with length, decreases with area, depends on the material, and increases with temperature for a metal
- A wire stretched to twice its length has four times the resistance, because the area halves as well
- **, with resistivity in ohm metre — a property of the material, not of the piece
- Metals and alloys** have resistivities of about to ohm metre; insulators about to
- Alloys are used for heating elements because of high resistivity and because they do not oxidise readily when hot
- Series: , the current is the same, the voltages add, and exceeds the largest resistor
- Parallel: , the voltage is the same, the currents add, and is below the smallest resistor
- Invert at the end of a parallel calculation, and check the magnitude
- Houses are wired in parallel so that every appliance gets the full voltage, can be switched independently, keeps working when another fails, and draws only the current it needs
- More appliances in parallel means less total resistance and more current — which is why overloading is a hazard
The sharpest self-test is the pair of checks. Take and ohms across V in parallel, find the branch currents, and confirm they add to the total; then put the same two in series and confirm the voltages add to V.
- Resistance increases with length, decreases with area, depends on the material, and increases with temperature for a metal
- A wire stretched to twice its length has four times the resistance, because the area halves as well
- **, with resistivity in ohm metre — a property of the material, not of the piece
- Metals and alloys** have resistivities of about to ohm metre; insulators about to
- Alloys are used for heating elements because of high resistivity and because they do not oxidise readily when hot
- Series: , the current is the same, the voltages add, and exceeds the largest resistor
- Parallel: , the voltage is the same, the currents add, and is below the smallest resistor
- Invert at the end of a parallel calculation, and check the magnitude
- Houses are wired in parallel so that every appliance gets the full voltage, can be switched independently, keeps working when another fails, and draws only the current it needs
- More appliances in parallel means less total resistance and more current — which is why overloading is a hazard
The sharpest self-test is the pair of checks. Take and ohms across V in parallel, find the branch currents, and confirm they add to the total; then put the same two in series and confirm the voltages add to V.