Two heater wires, made of the same material and having the same length and the same radius, are first connected in series and then in parallel to a constant potential difference. If the rate of heat produced in the two cases are Hs and Hp respectively then `(Hs)/(Hp)` will be

Options

A. `1//2`
B. 2
C. `1//4`
D. 4

Answer

C

Explanation

When heater wires are connected in series then equivalent resistance ,
`R_s = R_1 + R_2 = 2 R ( because R_(1) = R_(2))`
Rate of heat produced , `H_(x) = (V^(2))/(R_(x))`
`H_(s) = (V_(2))/(2 R) " " … (i)`
In second case , `R_(p) = (R_(1) R_(2))/(R_(1) + R_(2)) = (R * R)/(2 R)`
or `R_(p) = (R)/(2)`
`therefore ` Rate of heat produced , `H_(p)= (V^(2))/(R_(p))`
`H_(p) = (2V^(2))/(R) " " ... (ii)`
Dividing (i) by (ii) we get
`(H_(s))/(H_(p)) = ((V^(2)//2 R))/((2V^(2) // R)) = (v^(2))/(2R) xx (R)/(2V^(2)) = (1)/(4).`