Question

A uniform rectangular segment of copper wire with length and cross-sectional area has resistance: R. Then this resistor is melted down, and 1/5 the material is used to make a new uniform rectangular resistor with resistance R. What is the length of thisnew resistor?

Answer 1

Answer:

Explanation:

Let L and A be the length and area of cross section of rectangular segment

If resistor is melted down to and its $\frac{1}{5}$ material is used to make a new uniform rectangular resistor with same resistance R

volume of original rectangular(V)$=A\times L$

Volume of new rectangular segment with length $L_0(V_0)=A\times L_0$

$5V_0=V$

$5\times A_0\times L_0=A\times L$----1

Resistance is same

$R=\frac{\rho L}{A}=\frac{\rho L_0}{A_0}$

$\frac{L}{L_0}=\frac{A}{A_0}$

substitute $\frac{L}{L_0}$ in 1

$5=\frac{L^2}{L_0^2}$

$\frac{L}{L_0}=\sqrt{5}$

$L=\sqrt{5}L_0$

$L_0=\frac{L}{\sqrt{5}}$