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Electromagnetic Waves

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Physics Part I

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Physics

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A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?


Given, 
Frequency band of radio = 7.5 MHz - 12 MHz

 Wavelength, λ1 = cv1 = 3 × 1087.5 × 106 = 40 mWavelength, λ2 = cv2 = 3 × 10812 × 106 = 25 m 

Therefore, the corresponding wavelength band is 40 m - 25 m.

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Figure shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 mm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15A.
(a)    Calculate the capacitance and the rate of charge of potential difference between the plates.

(b)    Obtain the displacement current across the plates.

(c)    Is Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.


(a) Given,
Radius of capacitor plates, r = 12 cm = 0.12 m

distance between the plates, d = 5.0 mm = 5 × 10-3m

Charge carried, I = 0.15 A

Permittivity of medium, ε0 = 8.85 × 10-12 C2 N-1 m2

∴ Area of cross-section of plates, A = πR2 = 3.14 × (0.12)2 m

Capacitance of parallel plate capacitor is given by
                        C = ε0Ad         = 8.85 × 10-12× (3.14) × (0.12)25 × 10-3         = 80.1 × 10-12 = 80.1 pF

Now, charge on capacitor plate, q = CV 

                dqdt = C × dVdt

                     I = C × dVdt               I = dqdt 

                        dVdt = IC = 0.1580.1 × 10-12                      

                           = 1.87 × 109Vs-1 

(b) Displacement current is equal to the conduction current i.e., 0.15 A.

(c) Yes, Kirchhoff's first rule is valid at each plate of the capacitor provided. We take the current to be the sum of the conduction and displacement currents.

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What physical quantity is the same for X-rays of wavelength 10-10 m, red light of wavelength 6800 Å  and radio waves of wavelength 500 m?

The physical quantity speed is same for X-ray, red light and radio waves in vacuum and is c = 3 × 108 ms-1.
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A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?

Given, 
Plane EM wave travels in vacuum along z-direction. 
Frequency of the wave = 30 MHz 

E and B lie in x-y plane and are mutually perpendicular. 

Since, C =  we have, 
Wavelength of the EM wave, 

       λ = cv = 3 × 10830 × 106 = 10 m.
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A parallel plate capacitor (Fig.) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s-1.
(a)    What is the rms value of the conduction current?
(b)    Is the conduction current equal to the displacement current?
(c)    Determine the amplitude of B at a point 3.0 cm from the axis between the plates.



Given, a parallel plate capacitor made of circular plates.

(a) Here,
Radius of the circular plate, R = 6.0 cm
 Capacitance of the plates, C = 100 ρF = 100 × 10-12FAngular frequency, ω = 300 rad s-1

Erms = 230 V

Therefore, 

           Irms = ErmsXC = Erms1ωC = Erms × ωC 

        Irms  = 230 × 300 × 100 × 10-12
                 = 6.9 × 10-6A = 6.9 μA 

(b) Yes, the conduction current is equal to the displacement current.

I = ID,  whether conduction current, I is steady d.c. or a.c. This can be shown below :

          ID = ε0d(ϕE)dt = ε0 ddt(EA)       ϕE = EA

      ID = ε0AdEdt
            = ε0AddtQε0A       E = σε0 = Qε0A 
     ID = ε0A × 1ε0AdQdt    = dQdt = I
(c) We know that, 

                      B = μ02πrR2ID 

This formula goes through even if displacement current, ID (and therefore magnetic field B) oscillates in time. The formula above shows that they oscillate in phase.
Since I
D = I, we have
                      B = μ0rI2πR2 

If I = I0, the maximum value of current, then
Amplitude of B = maximum value of B 

= μ0rI02πR2  = μ0r2Irms2πR2                          I0 = 2 Irms= 4π × 10-7 × 0.03 × 2 × 6.9 × 10-62 × 3.14 × (0.06)2T= 1.63 × 10-11T.

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