Question

Two Cu wires of radii${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are such that (${\mathrm{R}}_1>{\mathrm{R}}_2$). Then which of the following is true?

• A. Transverse wave travels faster in thicker wire
• B. Transverse wave travels faster in thinner wire
• C. Travels with the same speed in both the wires
• D. Does not travel

Answer 1

The correct option is B

Transverse wave travels faster in thinner wire

Explanation for correct option:

(b) Transverse wave travels faster in thinner wire

Transverse waves:

1. A transverse wave is a wave whose oscillations are perpendicular to the wave's advance direction.
2. In contrast, a longitudinal wave moves in the direction of its oscillations.

Speed of transverse waves:

Speed of transverse wave in a wire is given as,

$\mathrm{v}=\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$where $\mathrm{\mu }$ is the mass per unit length of wire,
$\mathrm{\mu }=\frac{\mathrm{m}}{\mathrm{l}}=\frac{\mathrm{m}}{\mathrm{V}}\mathrm{A}=\mathrm{\rho A}$

Hence $\mathrm{v}=\sqrt{\frac{\mathrm{T}}{\mathrm{\rho A}}}$
$\mathrm{v}\propto \frac{1}{\sqrt{\mathrm{A}}}$

Thus, speed is higher for thinner wire (smaller area or smaller radius).

Explanation for incorrect option:

(a) Transverse wave travels faster in thicker wire

Speed of transverse wave is inversely proportional to the surface area of the wire

$\mathrm{v}\propto \frac{1}{\sqrt{\mathrm{A}}}$

So, this option is incorrect, as the speed will be lowered in this case.

(c) Travels with the same speed in both the wires

1. This option is incorrect.
2. Because clearly the speed depends on the surface area.

(d) Does not travel

1. As transverse waves can only travel through solids.
2. And Copper ($\mathrm{Cu}$) is a solid, so transverse waves will travel through copper wire.
3. So, this option is incorrect.

Hence, option (B) is the correct option.