10. Two metallic wires A and B are connected in parallel. Wire A has length L and radius r, wire B has a length 21 and
radius 2r, Compute the ratio of the total resistance of parallel combination and the resistance of wire A.
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Answer:
The electrical resistance of a uniform conductor is given in terms of resistivity by:
R=
a
ρl
where l is the length of the conductor in SI units of meters,
a is the cross-sectional area (for a round wire a=πr
2
if r is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters.
∴R
A
=
πr
2
ρl
and R
B
=
πr
2
2lρ
Resistance in parallel will be
R
eq
1
=
R
A
1
+
R
B
1
=
ρl
πr
2
+
2lρ
πr
2
=
ρ2l
3πr
2
∴R
eq
=
3πr
2
2ρl
Now the ratio between total resistance and reistance of wire A is
R
A
R
eq
=
πr
2
ρl
3πr
2
2ρl
=
3
2
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