Question

Two wires $A$ and $B$ of the same material and having the same length, have their cross sectional areas in the ratio \[1:6\] .What would be the ratio of heat produced in these wires when the same voltage is applied across each?

Answer 1

Hint:To answer this question, we must know the formula of heat produced and then comparing both the wires we will get the required ratio of heat produced in these wires when the same voltage is applied across each.

Formula used:
$H = {I^2}RT$
Where, $H$ is the heat produced, $I$ is the current, $R$ is the resistance and $T$ is the time.
$R = \dfrac{{\rho L}}{A}$
Where, $R$ is the resistance, $\rho $ is the resistivity of the material, $L$ is the length of the conductor and $A$ is the area of the cross section.

Complete step by step answer:
According to the question, it is given that the cross section area of wire A & B are in the ratio \[1:6\]. Now substituting the value of $R$ in the the formula of heat produced,
$\because H = {I^2}RT$
Heat produced in wire A,
${H_A} = {I^2} \times \dfrac{{\rho L}}{A} \times T \\
\Rightarrow {H_A} = {I^2} \times \dfrac{{\rho L}}{1} \times T \\ $
And, heat produced in wire B,
${H_B} = {I^2} \times \dfrac{{\rho L}}{A} \times T \\
\Rightarrow {H_B} = {I^2} \times \dfrac{{\rho L}}{6} \times T $
Since ${A_1};{A_2} = 1:6$ , the ratio of heat produced in wire A & B is:
${H_A}:{H_B} = {A_2}:{A_1} = 6:1$

Therefore, the ratio of heat produced in the wire A & B when the same voltage is applied across each wire is $6:1$.

Note:To solve this type of question one must remember the formula and remember while comparing the ratio here area is inversely proportional to heat produced so when we compare the ratio of heat produced when the voltage is same, the ratio we take is reciprocal and proceed accordingly.The given equation for heat produced is also called Joule’s equation of electrical heating.