Question

Two conductors made of the same material are connected across the same potential difference. Conductor A has seven times the diameter and seven times the length of conductor B. What is the ratio of the power delivere

Answer 1

Complete question:

Two conductors made of the same material are connected across the same potential difference. Conductor A has seven times the diameter and seven times the length of conductor B. What is the ratio of the power delivered to A to power delivered to B.

Answer:

The ratio of the power delivered to A to power delivered to B is 7 : 1

Explanation:

Cross sectional area of a wire is calculated as;

$A = \frac{\pi d^2}{4}$

Resistance of a wire is calculated as;

$R = \frac{\rho L}{A} \\\\R = \frac{4\rho L}{\pi d^2}$

Resistance in wire A;

$R = \frac{4\rho _AL_A}{\pi d_A^2}$

Resistance in wire B;

$R = \frac{4\rho _BL_B}{\pi d_B^2}$

Power delivered in wire;

$P = \frac{V^2}{R}$

Power delivered in wire A;

$P = \frac{V^2_A}{R_A}$

Power delivered in wire B;

$P = \frac{V^2_B}{R_B}$

Substitute in the value of R in Power delivered in wire A;

$P_A = \frac{V^2_A}{R_A} = \frac{V^2_A \pi d^2_A}{4 \rho_A L_A}$

Substitute in the value of R in Power delivered in wire B;

$P_B = \frac{V^2_B}{R_B} = \frac{V^2_B \pi d^2_B}{4 \rho_B L_B}$

Take the ratio of power delivered to A to power delivered to B;

$\frac{P_A}{P_B} = (\frac{V^2_A \pi d^2_A}{4\rho_AL_A} ) *(\frac{4\rho_BL_B}{V^2_B \pi d^2_B})\\\\ \frac{P_A}{P_B} = (\frac{V^2_A d^2_A}{\rho_AL_A} )*(\frac{\rho_BL_B}{V^2_B d^2_B})$

The wires are made of the same material, $\rho _A = \rho_B$

$\frac{P_A}{P_B} = (\frac{V^2_A d^2_A}{L_A} )*(\frac{L_B}{V^2_B d^2_B})$

The wires are connected across the same potential; $V_A = V_B$

$\frac{P_A}{P_B} = (\frac{ d^2_A}{L_A} )* (\frac{L_B}{d^2_B} )$

wire A has seven times the diameter and seven times the length of wire B;

$\frac{P_A}{P_B} = (\frac{ (7d_B)^2}{7L_B} )* (\frac{L_B}{d^2_B} )\\\\\frac{P_A}{P_B} = \frac{49d_B^2}{7L_B} *\frac{L_B}{d^2_B} \\\\\frac{P_A}{P_B} =\frac{49}{7} \\\\\frac{P_A}{P_B} = 7\\\\P_A : P_B = 7:1$

Therefore, the ratio of the power delivered to A to power delivered to B is

7 : 1