Question

A nichrome wire X with length (l) and cross-sectional area (A) is connected to a 10 V source and another nichrome wire Y with length (2l) and cross-sectional area `(A/2)`, is connected to a 20 V source.

(a) Compare the resistances of wires X and Y. [Given that the resistivity of nichrome is (ρ).]

(b) Compare the electrical power consumed by each wire.

(c) Compare the masses of these wires. (Given that the density of nichrome is d.)

(d) State true or false:

Wire X and wire Y both show the same rise in temperature in the same time.

Answer 1

Given: Length of nichrome wire X (l_X) = l

Cross-sectional area of nichrome wire X (A_X) = A

Potential difference across nichrome wire X (V_X) = 10 V

Length of nichrome wire Y (l_Y) = 2l

Cross-sectional area of nichrome wire Y (A_Y) = `A/2`

Potential difference across nichrome wire Y (V_Y) = 20 V

resistivity of nichrome = ρ

(a) Let ‘R_X’ and ‘R_Y’ be the resistance of nichrome wires X and Y, respectively.

Now, R_X​ = `ρ l/A`

R_Y​ = `ρ (2l)/(A/2)`

= `ρ (4l)/A`

= 4R_X

As, `R_Y/R_X` = 4

So, R_X : R_Y = 1 : 4

(b) P_X = `((V_X)^2)/R_X`

= `10^2/R_X`

= `100/R_X`

P_Y = `((V_Y)^2)/R_Y`

= `20^2/(4R_X)`

= `400/(4R_X)`

= `100/R_X`

= P_X

So, P_X : P_Y = 1 : 1

(c) Given, mass density of nichrome = d

Then, mass of wire X (m_X) = d × volume = d × Al

Mass of wire Y (m_Y) = d × volume

= `d xx 2l xx A/2`

= d × Al

= m_X

So, m_X : m_Y = 1 : 1

(d) This statement is true.

Explanation:

Let the heat gained by the wire X be ‘Q_X’ and the change in temperature be ‘ΔT_X’. Similarly, heat gained by the wire Y is ‘Q_Y’ and the change in temperature is ‘ΔT_Y’.

Again, let the specific heat capacity of nichrome be ‘c’ and at the same time be ‘t’.

Then, Q_X = P_X × t

= m_Xc × ΔT_X

And Q_Y = P_Y × t

= m_Yc × ΔT_Y

As, P_X = P_Y then Q_X = Q_Y for same time t.

⇒ m_Xc × ΔT_X = m_Yc × ΔT_Y

⇒ m_X × ΔT_X = m_Y × ΔT_Y

So, ΔT_X : ΔT_Y = m_X : m_Y = 1 : 1

Hence, wire X and wire Y both show the same rise in temperature in the same time.