Question

4.1 The ratio of the lengths of two wires is 2 : 1 and the ratio of their radii is 1:2. Calculate the ratio of the resistance of the two wires if their resistivity is in a 3 : 4 ratio.​

Answer 1

Answer:

R

1

=(

ρ

2

ρ

1

)(

L

2

L

1

)(

A

1

A

2

)

=(

ρ

2

ρ

1

)(

L

2

L

1

)(

r

1

2

r

2

2

)

=

3

2

×

5

3

×(

7

2

)

−2

[

r

1

r

2

=

2

7

]

=

5

2

×(

49

4

)

−2

=

5

2

×

4

49

R

2

R

1

=

10

49

Answer 2

Given :  Ratio of length of two wires is 2:1

             Ration of radii of two wires is 1:2

             Resistivity of two wires is 3:4

To Find : Ratio of Resistance of two wires

Solution :

Here, Resistance of a wire with length L , cross-sectional area A of radius R and resistivity ρ  is given by

R = $\[\rho \frac{L}{A}\]$

Let lengths, radii and resistivity of wire1 and wire2 be $\[{{L_1}}\]$, $\[{{L_2}}\]$, $\[{{r_1}}\]$, $\[{{r_2}}\]$ and $\[{{\rho _1}}\]$, $\[{{\rho _2}}\]$ respectively

Thus .

$\[\frac{{{R_1}}}{{{R_2}}} = \frac{{{\rho _1}}}{{{\rho _2}}} \times \frac{{{L_1}}}{{{L_2}}} \times \frac{{{A_2}}}{{{A_1}}}\]$

$\[\frac{{{R_1}}}{{{R_2}}} = \frac{3}{4} \times \frac{2}{1} \times \frac{{\pi {r_2}^2}}{{\pi {r_1}^2}}\]$

$\[\frac{{{R_1}}}{{{R_2}}} = \frac{3}{4} \times \frac{2}{1} \times \frac{{{2^2}}}{{{1^2}}}\]$

$\[\frac{{{R_1}}}{{{R_2}}} = \frac{3}{4} \times \frac{2}{1} \times \frac{4}{1}\]$

$\[\frac{{{R_1}}}{{{R_2}}} = \frac{6}{1}\]$

Hence, the ratio of resistance of wire1 and wire2 is 6:1.