Two wires A and B made of the same material and having the same lengths are connected across the same voltage source. If the power supplied to wire A is six times the power supplied to wire B, what is the ratio of their diameters

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)