Question

A circular wire loop has a radius of 7.50 cm. a sinusoidal electromagnetic plane wave traveling in air passes throughthe loop, with the direction of the magnetic field of the wave perpendicular to the plane of the loop. the intensity of the wave atthe location of the loop is 0.0275 w>m2, and the wavelength of thewave is 6.90 m. what is the maximum emf induced in the loop

Answer 1

I = intensity of the wave at the location of the loop = 0.0275 W/m²

B₀ = amplitude of magnetic field = ?

Intensity is given as

I = B²₀ c/(2μ₀)

inserting the values

0.0275 = B²₀ (3 x 10⁸)/(2(12.56 x 10⁻⁷))

B₀ = 1.52 x 10⁻⁸ T

λ = wavelength of the wave = 6.90 m

frequency of the wave is given as

f = c/λ = (3 x 10⁸)/(6.90) = 4.35 x 10⁷ Hz

angular frequency is given as

w = 2πf = 2 x 3.14 x 4.35 x 10⁷ = 2.73 x 10⁸ rad/s

r = radius of the loop = 7.50 cm = 0.075 m

A = area of the loop = πr² = (3.14) (0.075)² = 0.0177 m²

maximum induced emf is given as

E = B₀ A w

inserting the values

E = (1.52 x 10⁻⁸) (0.0177) (2.73 x 10⁸)

E = 0.0735 Volts

E = 73.5 mV

Answer 2

Intensity of electromagnetic wave is given as

`I = 2[(B_(rms)^2)/(2\mu_0)* c]`

given that

`I = 0.0275 W/m^2`

`0.0275 = (B_(rms)^2)/(\mu_0) * c`

here we know that

`\mu_0 = 4\pi * 10^(-7)`

`c = 3 * 10^8 m/s`

now we have

`0.0275 = (B_(rms)^2)/(4 \pi * 10^(-7))(3 * 10^8)`

`B_(rms) = 1.07 * 10^(-8) T`

now we will have

`B_0 = \sqrt2 B_(rms)`

`B_0 = 1.52 * 10^(-8) T`

frequency of wave is given as

`f = (c)/(\lambda)`

`f = (3 * 10^8)/(6.90) = 4.35 * 10^7 Hz`

now the induced EMF is given as

`EMF = B_0A2\pi f`

`EMF = 1.52 * 10^(-8) * \pi(0.075)^2 * (2\pi * 4.25 * 10^7)`

`EMF = 0.0733 Volts`