Question

A wire of radius r and another wire of radius \[2r\], both of the same material and length are connected in series to each other. The combination is connected across a battery. The ratio of the heat produced in the two wires will be
A. \[4.00\]
B. \[2.00\]
C. \[0.50\]
D. \[0.25\]

Answer 1

Hint:The resistance R is proportional to the heat generated in the circuit. Determine the equivalent resistance of the wires in series connections. Use the electrical power formula (heat produced). When there are two or more sections through which current can flow, the circuit is said to be parallel.

Formula used:
\[H = {I^2}Rt\] describes the heating effect caused by an electric current, I, flowing through a conductor of resistance, R, for a period of time, t. Joule's equation of electrical heating is the name of this formula.
The formula is \[H = {I^2}Rt\]

Complete step by step solution:
We have been given in the question that a wire of radius \[{\rm{r}}\] and another wire of radius \[2{\rm{r}}\] both of same material and length are connected in series to each other. The combination is connected across a battery. We have to determine the ratio of the heats produced in the two wires. We already know the formula of Heat produced,
\[{\rm{H}} = {1^2}{\rm{Rt}}\]
Here, current will be the same, because both wires are connected in series.
Thus, \[{\rm{H}} \propto {\rm{R}}\]

Now, let us have
\[R = \dfrac{{\rho l}}{{\pi {r^2}}}\]
Here \[1,\rho \] are the same for both, because both wires have the same length and same material.
Thus from the previous calculation, we get
\[{\rm{R}} \propto \dfrac{1}{{{{\rm{r}}^2}}}\]
As the combination is across and connected in series, we have
\[\dfrac{{{{\rm{R}}_1}}}{{{{\rm{R}}_2}}} = \dfrac{{{{(2{\rm{r}})}^2}}}{{\rm{r}}}\]

Now, we have to substitute the value of radius of wire in the formula, we get
\[\dfrac{{{{\rm{R}}_1}}}{{{{\rm{R}}_2}}} = 4\]
And hence we obtain,
\[\dfrac{{{{\rm{H}}_1}}}{{{{\rm{H}}_2}}} = \dfrac{{{{\rm{R}}_1}}}{{{{\rm{R}}_2}}} = 4\]
Therefore, the ratio of the heats produced in the two wires will be \[4\].

Hence, option A is correct.

Note: Electrical power is the rate at which a circuit consumes electric energy. A series circuit has only one current flow path. Remember that when computing equivalent resistance for parallel circuits, it is always an inverse first. Resistance is the same as the wire's construction. Electrical power is a type of heat production as well.