two wires,one of copper and other of manganin, have equal lengths and equal resistances.Which wire is thicker ?Given that resistivity of copper is lower than that of manganin

as resistivity of copper is lower than that of manganin it will be thicker . as the thickker the wire the less resistance will it provide

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**R ∝ 1/A where R = resistance and A = cross sectional area of wire, or thickness of wire**

**R ∝ ρ where R resistance and **

**ρ = resistivity**=> **ρ ****∝ 1/A where ****ρ = resistivity ****A = cross sectional area of wire, or thickness of wire**

Given = Length and resistance of copper and manganin are equal and resistivity of copper is less than that of manganin

Taking **ρ ****∝ 1/A** we can say that copper is more thicker than manganin, as copper's resistivity is lesser compared to that of manganin.

Hope this helps!!

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R=rho x L/A

where R= resistance of conductor

L=length of conductor

A= area of cross section of the conductor

for Copper----------R_{Cu}= rho_{Cu} x L_{Cu}/A_{Cu}

for Manganin------ R_{Manganin}= rho _{Ma}_{ngnanin} x L_{Manganin} /A_{Manganin}

but R_{Cu}= R_{manganin} (given),

and L_{Cu}= L_{manganin} (given),

Since R_{Cu}= R_{manganin}

therefore rho _{Cu} x L_{Cu}/A _{Cu} = rho _{manganin} x L _{manganin}/ A _{manganin}

Cancelling L_{Cu} and L _{manganin} of LHS and RHS respectively, we get,

rho _{Cu}/ A_{CU}= rho _{manganin} /A _{manganin},

but rho of copper is less than rho of manganin, (given),

(since the LHS numerator is less than the RHS numerator),

but LHS = RHS

therefore the LHS denominator is less than the denominator of RHS,

ie. the A _{CU} is less than the A _{manganin}.(denominators)

Therefore the area of cross section of the Copper wire is less the area of cross section of the manganin wire

ie. the copper wire is thicker than the manganin wire.

PROVED.

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**Manginin wire is thicker.**

it is given that copper has less resistivity than manganin. That means copper wire will have less resistance than manganin wire as it is given that they have equal lengths. But it is also given that they have equal resistance. To prove this manganin wire must be thicker than copper wire as

**RESISTANCE IS INVERSELY PROPORTIONAL TO THE AREA OF CROSS-SECTION**which means more the area, less the resistance. THEREFORE MANGANIN WIRE WITH EQUAL RESISTANCE AND EQUAL LENGTH AS A COPPER WIRE WILL BE THICKER THAN THE COPPER WIRE. If u understand plz give a thumbs up :)

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**Manginin wire is thicker.**

it is given that copper has less resistivity than manganin. That means copper wire will have less resistance than manganin wire as it is given that they have equal lengths. But it is also given that they have equal resistance. To prove this manganin wire must be thicker than copper wire as

**RESISTANCE IS INVERSELY PROPORTIONAL TO THE AREA OF CROSS-SECTION**which means more the area, less the resistance. THEREFORE MANGANIN WIRE WITH EQUAL RESISTANCE AND EQUAL LENGTH AS A COPPER WIRE WILL BE THICKER THAN THE COPPER WIRE. If u understand plz give a thumbs up :)

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Thank you for giving such a detailed and well-written proof! :) :)

However, I'd like to make one little correction.

In the last line, you've written, "Therefore, the area of cross section of the copper wire is less than the area of cross section of the manganin wire", which is right.

But, the next line contradicts your statement as it is written "i.e., Copper wire is thicker than the Manganin wire", which should have been written as "ie.,

**Manganin wire is thicker than Copper wire**".

Thank you for the answer!

-Akhila

- -3

Which means that,

Rho = RA/l,

where Rho (resistivity) is directly proportional to A (area of cross-section).

- We are given that Rho (Resistivity) of Copper is lesser than Rho (Resistivity) of Manganin.
- Hence, in the case of Copper, since resistivity is less, area of cross section is also less. (Rho is directly proportional to A) Hence, it is thinner.
- In the case of Manganin, since resistivity is comparitively more, area of cross-section will also be comparitively more. (Rho is directly proportional to A) Hence, it is thicker.
- So,
**Manganin wire is thicker than Copper wire**.

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