Two heating coils, one of fine wire and the other of thick wire made of same material and of same length are first connected in series and then in parallel. Which of the following statements is correct?

# Two heating coils, one of fine wire and the other of thick wire made of same material and of same length are first connected in series and then in parallel. Which of the following statements is correct?

1. A

In series fine wire liberates more energy while in parallel thick wire liberates more energy.

2. B

In series fine wire liberates less energy while in parallel thick wire liberates less energy.

3. C

In series thick wire liberates more energy while in parallel it liberates less energy.

4. D

Both wires liberates equal energies in series and in parallel.

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### Solution:

${\mathrm{R}}_{1}=\frac{\mathrm{\rho l}}{{\mathrm{A}}_{1}},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{R}}_{2}=\frac{\mathrm{\rho l}}{{\mathrm{A}}_{2}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{As}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{A}}_{1}\text{\hspace{0.17em}\hspace{0.17em}}<\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{A}}_{2}\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{so}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{R}}_{1}\text{\hspace{0.17em}}>\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{R}}_{2}$

In series $\mathrm{H}={\mathrm{I}}^{2}\mathrm{Rt}\text{\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{H}\text{\hspace{0.17em}}\propto \text{\hspace{0.17em}\hspace{0.17em}}\mathrm{R}\text{\hspace{0.17em}\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{H}}_{1}\text{\hspace{0.17em}\hspace{0.17em}}>\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{H}}_{2}$

In parallel $\mathrm{H}=\frac{{\mathrm{V}}^{2}}{\mathrm{R}}\mathrm{t}\text{\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{H}\text{\hspace{0.17em}}\propto \text{\hspace{0.17em}\hspace{0.17em}}\frac{1}{\mathrm{R}}\text{\hspace{0.17em}\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{H}}_{1}\text{\hspace{0.17em}\hspace{0.17em}}<\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{H}}_{2}$

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