Today we will discuss the multiple choice questions on waves included in All India Pre-Medical / Pre-Dental Entrance Examination (AIPMT) 2009 and 2008 question papers. Here are the AIPMT 2009 questions:

(1) The electric field part of of an electromagnetic wave in a medium is represented by

E_{x} = 0;

E_{y} = 2.5 N/C cos[(2π×10^{6} rad/s)t – (π×10^{–2} rad/m)x];

E_{z} = 0.

The wave is:

(1) moving along x-direction with frequency 10^{6} Hz and wave length 100 m.

(2) moving along x-direction with frequency 10^{6} Hz and wave length 200 m.

(3) moving along –x-direction with frequency 10^{6} Hz and wave length 200 m.

(4) moving along y-direction with frequency 2π×10^{6} Hz and wave length 200 m.

The electric field variation is in accordance with the equation* y = **A* sin (*ωt – kx***)** which represents a progressive wave proceeding along the positive x-direction. Instead of the usual displacement *y* we have the electric field E. In place of the angular frequency *ω* we have 2π×10^{6} (remember *ω = *2π*n*) which means that the linear frequency *n* is 10^{6} Hz.

Since the propagation constant *k* = 2π/*λ* where *λ* is the wave length, we have

π×10^{–2} = 2π/*λ* from which *λ = *200 ms^{–1}.

So the correct option is (2).

[The units of electric field E (N/C), angular frequency* ω* (rad/s) and the propagation constant *k* (rad/m) given in the wave equation in the question should not distract you.

You should note that the *negative* sign in the wave equation *y = **A* sin (*ωt – kx*) or *y = **A* sin (*kx –** ωt*) indicates that the wave is propagating along the* positive* x-direction. A wave propagating along the* negative* x-direction is represented by *y = **A* sin (*ωt + kx*).

You can make use of any other form of the wave equation, for instance, *y =** A* sin [2**π( t/T – x/**

*λ***)],**also to solve the problem].

**(2)** A wave in a string has an amplitude of 2 cm. The wave travels in the + ve direction of x axis with a speed of 128 m/sec. and it is noted that 5 complete waves fit in 4 m length of the string. The equation describing the wave is:

(1) *y* = (0.02) m sin (15.7*x* − 2010*t*)

(2) *y *= (0.02) m sin (15.7*x* + 2010*t*)

(3) *y* = (0.02) m sin (7.85*x *− 1005*t*)

(4) *y* = (0.02) m sin (7.85*x *+ 1005*t*)

The unit of amplitude is metre, shown by its symbol ‘m’ in the equation (which should not distract you).

The equation describing the wave propagating in the +ve x-direction is

*y = **A* sin (*kx –** ωt*)

The amplitude *A* as given in the question is 2 cm = 0.02 m.

The velocity of the wave, *v =** ω/k = *128 ms^{–1}. * *

Wave length *λ* = 4/5 m.

But *k** = *2π/*λ = *2π×5/4 = **7.85** m and so *ω** = kv = *7.85×128 = **1005**

So the equation of the wave is

*y* = (0.02) m sin (7.85*x *− 1005*t*)

The correct option is (3).

Here is the AIPMT 2008 question:

The wave described by *y = *0.25 sin(10π*x* – 2π*t*) where *x* and *y* are in metres and *t* in seconds is a wave traveling along the

(1) negative *x*-direction with amplitude 0.25 m and wavelength *λ* = 0.2 m.

(2) negative *x*-direction with frequency 1 Hz.

(3) positive *x-*direction with frequency π Hz and wavelength *λ* = 0.2 m.

(4) positive *x-*direction with frequency 1 Hz and wavelength *λ* = 0.2 m.

The given wave equation is in the form *y = A* sin [2π(*x/**λ – **t/T*)].

The *negative* sign in the equation shows that the wave is propagating along the* positive* x-direction.

By comparison we obtain 2π*x/**λ* = 10π*x* from which *λ* = 0.2 m. Also, 2π*t/T =*2π*t** *from which the frequency 1/T = 1 Hz. [Option (4)].

You will find a useful post on waves at AP Physics Resources.