Question

7.Two wires A and B are of equal lengths, different cross-sectional areas and made of the same metal.
(i) Name the property which is same for both the wires.
(ii) Name the property which is different for both the wires.
(iii) If the resistance of wire A is four times the resistance of wire B, calculate: (a)the ratio of the cross-
sectionai areas of the wires and (b)the ratio of the radii of the wires.​

Answer 1

Answer:

Given:

Wires A and B are of equal length , different Cross-sectional areas but of same material.

i) Property which is same for both the wires is Resistivity. Resistivity is a materialistic property which is independent of dimensions (like length or breadth)

ii) The property which is different for both the wires will be Resistance.

Resistance depends upon :

• Length of wire
• Area of cross-section
• Resistivity of material.

Simce the wires have different area of cross-section , tge resistance will also be different.

iii) Let resistance of A be R1 and that of B be R2.

$R1 = 4 \: \times (R2)$

$= > \rho( \frac{l1}{a1} ) = 4 \{ \rho( \frac{l2}{a2} ) \}$

Cancelling Length on both sides :

$= > \frac{1}{a1} = 4 ( \frac{1}{a2} )$

$= > \frac{a1}{a2} = \frac{1}{4}$

Therefore a1 : a2 = 1 : 4

$= > \frac{\pi {(r1)}^{2} }{\pi {(r2)}^{2} } = \frac{1}{4}$

$= > \frac{r1}{r2} = \frac{1}{2}$

Therefore r1 : r2 = 1 : 2

Answer 2

SoluTion :

✴ Given :-

▪ Two wires A and B are of equal lengths, different coss-sectional areas and made of the same metal.

1) The property which is same for both the wires is resistivity.

▪ Resistivity is materialistic property.

▪ It doesn't depends upon length or area of conductor.

▪ Here, both wires are made up from same material therefore resistivity of both will be same.

2) The property which is not same for both the wires is resistance.

▪ Magnitude of resistance is directly proportional to the length of the conductor and inversely proportional to the area of cross-section.

3) Resistance of wire A is four times the resistance of wire B

• Ratio of area of cross-section

$\implies\sf\:\rho_A=\rho_B\\ \\ \implies\sf\:\dfrac{R_A\cancel{l}}{A_A}=\dfrac{R_B\cancel{l}}{A_B}\\ \\ \implies\sf\:\dfrac{A_A}{A_B}=\dfrac{R_B}{R_A}\\ \\ \implies\:\boxed{\sf{\large{\pink{A_A:A_B=1:4}}}}$

• Ratio of the radiii of the wires

$\implies\sf\:\dfrac{A_A}{A_B}=\dfrac{\pi{r_A}^2}{\pi{r_B}^2}\\ \\ \implies\sf\:\dfrac{{r_A}^2}{{r_B}^2}=\dfrac{1}{4}\\ \\\implies\:\boxed{\sf{\purple{\large{r_A:r_B=1:2}}}}$