Question

i) Wave motion is doubly periodic. Explain.
ii) Four coulomb charge is uniformly distributed on 2 km long wire. Calculate its linear charge density.
iii) Name a constant which is dimensionless.

Answer 1

Hint: Doubly periodic means it is periodic in time and periodic in space. The linear charge density is the ratio of charge on the conductor to the length of the conductor. A dimensionless quantity is unit less.

Formula used:
Linear charge density, \[\lambda = \dfrac{Q}{L}\]
Here, Q is the charge on the wire and L is the length of the wire.

Complete step by step answer:

(i) We know that the time taken by a wave to complete one revolution is known as the time period of the wave. Therefore, we can say that the wave repeats itself after a certain interval of time. So, the wave motion is periodic in time.
The distance between the two consecutive points of the wave performing oscillations when the wave is said to be in phase always remains the same. This distance between these two points we call the wavelength of the wave. Since the wavelength of the wave repeats itself in the same manner as that of the previous one, we can say that the wave motion is periodic in space.
Therefore, the wave motion is doubly periodic: periodic in time and periodic in space.

ii)We have the formula for linear charge density,
\[\lambda = \dfrac{Q}{L}\]
Here, Q is the charge on the wire and L is the length of the wire.
Substituting 4 C for Q and 2 km for L in the above equation, we get,
\[\lambda = \dfrac{4}{{2 \times {{10}^3}}}\]
\[ \Rightarrow \lambda = 2 \times {10^{ - 3}}\,C/m\]
Therefore, the linear charge density of the wire is \[2 \times {10^{ - 3}}\,C/m\].

iii)We know the constant Reynolds number and Avogadro’s number. These constants have no unit and therefore dimensionless.

Note:
To answer the first part, students should remember the period and wavelength of the wave.
The linear charge density has unit C/m.
Therefore, make sure that you have converted the length of wire into meters.