Answer:
sqrt (3) : sqrt (2)
Explanation:
Given that Two wires A and B made of the same material and having the same lengths are connected across the same voltage source.
Since they are both made of the same material, they will have the same resistivity.
Resistance is directly proportional to the length of the wire and inversely proportional to the cross sectional area.
Resistance R =1/ πr^2
If the power supplied to wire A is six times the power supplied to wire B, that is,
Pa = 6Pb
Where
Power P = IV
Since the same voltage passes through both wires,
P = V^2/R
Substitutes resistance R into the equation .Therefore,
V^2 ÷ 1 / πr^2 = 6( V^2 ÷ 1 / πr^2 )
V will cancel out , leaving
πr^2 = 6πr^2
Pi( π) will also cancel out. And since diameter d = 2r
r = d/2
Substitutes r into the equation
(d/2)^2 = 6( d/2)^2
d^2/4 = 6( d^2/4)
d^2/4 = 3d^2/2
da^2/2 = 3db^2
(Da/Db)^2 = 3/2
Da : Db = sqrt (3) : sqrt (2)
therefore, the ratio of their diameters is
sqrt (3) : sqrt (2)
The ratio of the diameter of Wire A to Wire B is; √6 : 1
Since both wires are made of the same material, they will definitely have the same resistivity.
Formula for the resistance with area is;
R = 1/A = 1/(πr²)
We are told that the power supplied to wire A is six times the power supplied to wire B. Thus;,
Pa = 6Pb
Since the same voltage passes through both wires, then formula for power in both cases is; P = V²/R. Thus;
V² ÷ (1/πr_a²) = 6(V² ÷ (1/πr_b²))
V and π will cancel out to give;
r_a² = 6r_b²
We know that radius; r = d/2
Thus;
(d_a/2)²= 6(d_b/2)²
d_a²/4 = 6(d_b²/4)
(d_a/d_b)² = 6
d_a/d_b = √6 : 1
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