The electric fields of two microwaves traveling in opposite directions are given

Question

The electric fields of two microwaves traveling in opposite directions are given by: E_1 (x, t) = 7sin3 pi (2 times 10^2 x - 4 times 10^10t)^- E_2 (x, t) = 7sin3 pi (2 times 10^2 x + 4 times 10^10 t) a. Calculate the frequency and the wavelength for each wave. b. Determine the speed and the direction of propagation for each wave. c. Write the wave function of the standing wave that E_1 and E_2 produce. d. Determine the location of the nodes having the third and the fourth smallest value of x (x > 0).

Explanation / Answer

common form of wave equation:

E(x,t)=A*sin(k*x-w*t)

where A=amplitude

k=wave number=2*pi/wavelength

==>wavelength=2*pi/k

w=angular frequency=2*pi*frequency

==>frequency=w/(2*pi)

part a:

for first wave:

k=3*pi*2*10^2

==>wavelength=2*pi/k=3.333 mm

w=3*pi*4*10^10

frequency=w/(2*pi)

=6*10^10 Hz

for second wave:

k=3*pi*2*10^2

wavelength=2*pi/k

=3.333 mm

w=3*pi*4*10^10

frequency=w/(2*pi)

=6*10^10 Hz

part b:

as for both the waves, wavelength and frequency are same,

speed=wavelength*frequency=2*10^8 m/s

direction of E1 is along +ve x direction

and direction of E2 is along -ve x direction

part c:

standing wave=E=E1+E2

=7*sin(3*pi*(2*10^2*x-4*10^10*t)) + 7*sin(3*pi*(2*10^2*x+4*10^10*t))

using the formula:

sin(A+B)+sin(A-B)=2*sin(A)*cos(B)

E=7*2*sin(3*pi*2*10^2*x)*cos(3*pi*4*10^10*t)

part d:

nodes will occur where amplitude=0

amplitude=14*sin(3*pi*2*10^2*x)

it will be zero, when 3*pi*2*10^2*x=n*pi

where n can be any integer

as given condition is x>0, n=1,2,3,....

for third and fourth smallest value of x,

n=3 and n=4

using n=3:

x=3*pi/(3*pi*2*10^2)=0.005 m

using n=4:

x=4*pi/(3*pi*2*10^2)=0.00667 m

hence third and fourth smallest values of location of a node is 0.005 m and 0.00667 m

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.