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H C VERMA PHYSICS BOOK SOLUTIONS ELECTROMAGNETIC WAVES

Solved Examples

Ex1.    A parallel-plate capacitor is being charged. Show that the displacement current across an area in the region between the plates and parallel to it is equal to the conduction current in the connecting wires.
                                
Sol.    The electric field between the plates is 
          
    where Q is the charge accumulated at the positive plate. 
    The flux of this field through the given area is
          
    The displacement current is 
          
    But dQ/dt is the rate at which the charge is carried to the positive plate through the connecting wire. Thus , id = ie

Ex2.    The maximum electric field in a plane electromagnetic wave is 600 N/C . The wave is going in the x-direction and the elelctric field is in the y-direction. Find the maximum magnetic field in the wave and its direction.
Sol.    We have
      
    As ,  and the direction of propagation are mutually perpendicular,  should be along the z-direction.

Ex3.    The electric field in an electromagnetic wave is given by 
        E = (50 N/C) sinw (t – x/c).
    Find the energy contained in a cylinder of cross-section 10 cm2 and length 50 cm along the X-axis.
Sol.    The energy density is
      

Ex4.    Find the intensity of the wave discussed in example (40.3) ?
Sol.    The intensity is 
      

Questions for Short answer

1.    In a microwave oven , the food is kept in a plastic container and the microwave is directed towards the food. The food is cooked without melting or igniting the plastic container. Explain.

2.    A metal  rod is placed along the axis of a solenoid carrying a high-frequency alternating current. It is found that the rod gets heated. Explain why the rod gets heated.

3.    Can an electromagnetic wave be deflected by an electric field ? By a magnetic field ?

4.    A wire carries an alternating current i = i0 sinwt . Is there an electric field in the vicinity of the wire ?

5.    A capacitor is connected to an alternating-current source. Is there a magnetic field between the plates?

6.    Can an electromagnetic wave be polarized ?

7.    A plane electromagnetic waves is passing through a region. Consider the quantities (a) electric field , (b) magnetic field , (c) electrical energy in a small volume and (d) magnetic energy in a small volume . Construct pairs of the quantities that oscillate with equal frequencies.

Objective - I

1.    A magnetic field can be produced by - 
    (A) a moving charge                            (B) a changing electric field 
    (C) none of them                            (D*) both of them  
    

2.    A compass needle is placed in the gap of a parallel plate capacitor. The capacitor is connected to a battery through a resistance. The compass needle 
    (A) does not delfect
    (B) deflects for a very short time and then comes back to the original position    
    (C) deflects and remains deflecred as long as the battery is connected    
    (D*) deflects and gradually comes to the original position in a time which is large compared to the         time constant.  
     

3.    Dimensions of 1/(m0e0) is  
    
        (A) L/T                (B) T/L                    (C*) L2/T2                (D) T2/L2

4.    Electromagnetic waves are produced by 
    (A) a static charge                        (B) a moving charge        
    (C*) an accleration charge                (D) chargeless particles  
    
5.    An electromagnetic wave going through vacuum is described by
      
    
6.    An electric field and a magnetic field exist in a region. The fields are not perpandicular to each other. 
    (A)     This is not possible        
    (B) No electromagnetic wave may be passing through the region
    (C*) An electromagnetic wave may be passing through the region.
    (D) An electromagnetic wave is certainly passing through the region  
    

7.    Consider the following two statement regarding a linearly polarized, plane electromagnetic wave
    (A) The electric field and the magnetic field have equal average values
    (B) The electric energy and the magnetic energy have equal average values
    (A*) Both A and B are true            (B) A is false but B is true   
    (C) B is false but A is true            (D) Both A and B are false   
    
8.    A free electron is placed in the path of a plane electromagnetic wave. The electron will start moving 
    (A*) along the electric field            
    (B) along the magnetic field
    (C) along the direction of propagation of the wave
    (D) in a plane containing the magnetic field and the direction of propagation  
    

9.    A plane ekectromagnetic wave is incident on a material surface. The wave delivers momentum p and energy E.  
    
  

Objective - II

1.    An electromagnetic wave going through vacuum is described by E = E0 sin(kx - wt). Which of the following is/are independent of the wavelength ?  
    
    (A) k                (B) w                    (C*) k/w                        (D) kw

2.    Displacement current goes through the gap between the plates of a capacitor when the charge of the capacitor
    (A*) increases        (B*) decreases            (C) does not change            (D) is zero  
    

3.    Speed of electromagnetic waves is the same
    (A) for all wavelength        (B)     in all media        (C*) for all intensities    (D) for all frequencies  
    
4.    Which of the following have zero average value in a plane electromagnetic wave ?
    (A*) electric field        (B*) magnetic field    (C) electric energy    (D) magnetic energy  
    

5.    The energy contained in a small volume through which an electromagnetic wave is passing oscillates with
    (A) zero frequency                            (B) the frequency of the wave     
    (C) half the frequency of the wave                (D*) double the frequency of the wave  
    
Worked Out Examples

1.    A parallel-plate capacitor with plate area A and separation between the plates d, is charged by a constant current i. Consider a plane surface of area A/2 parallel to the plates and drawn symmetrically between the plates. Find the displacement current through this area.
Sol.    Suppose the charge on the capacitor at time t is Q. The electric field between the plates of the capacitor

2.    A plane electromagnetic wave propagating in the x-direction has a wavelength of 5.0 mm. The electric field is in the y-direction and its maximum magnitude is 30 V/m. Write suitable equations for the electric fields in the wave may be written as 
        
 
    
3.    A light beam travelling in the x-direction is described by the eletric field Ey = (300 V/m) sinw(t – x/c). An electron is constrained to move along the y-direction with a speed of 2.0 × 107 m/s. Find the maximum electric force and the maximum magnetic force on the electron.
Sol.    The maximum electric field is E0 = 300 V/m. The maximum magnetic field is 
          

4.    Find the energy stored in a 60 cm length of a laser beam operating at 4 mW.
Sol. 
                
    The time taken by the electaromagnetic wave to move through a distnce of 60 cm is t = 60cm/c= 2 × 10–9 s.The energy contained in the 60 cm length passes through a cross-section of the beam in 2×10–9 s. But the energy passing through any cross-section in 2 ×10–9 s is 
              
    This is the energy contained in 60 cm length.

5.    Find the amplitude of the electric field in a parallel beam of light of intensity 2.0 W/m2. 
Sol.    The intensity of a plane electromagnetic wave is 
             
        

Exercise

Q.1    A point charge is moving along a straight line with a constant velocity v consider a small area A perpendicular to the direction of motion of the charge (figure 40= E1) Calculate the displacement current through the area when its distance from the charge is x .The value of x is not large so that the electronic given by coulomb’s field at any instant is essentially given by Coulomb’s law.  
    
      

Q.2    A parallel-plate capacitor having plate area A and plate separation d is joined to a battery of emf e and internal resistance R at t = 0 consider a plane surface of area A/2 parallel to the plates and situated symmetrically between them. Find the displacement current through this surface as a function of time.
      

Q.3    Consider the situation of the previous problem Define displacement resistance Rd = V/ id of the space between the plates where V is the potential difference between the plates and is the displacement currently show that Rd varies with time as 
      
    
Q.4    Using B = µ0 H find the ratio Eo/ Ho for a plane electromagnetic wave propagating through vacuum. Show that it has the dimensions of electric resistance This ratio is a universal constant called the impedance of free space.  
     
Ans:    377 W

Q.5    The sunlight reaching the earth has maximum electric field of 810 V/m .What is the maximum magnetic field in this light ?   
      
Ans:     2.7 µ T  

Q.6    The magnetic field in a plane electromagnetic wave is given by 
    B =( 200 µ T ) sin [(4.0 × 10 15 s– 1) (t – x /c].    
    Find the maximum electric field and the average energy density corresponding to the electric field  
    
Ans:    6 × 10 4 N/C , 0.016 J/m 3  

Q.7    A laser beam has intensity 2.5 × 20 14 W/m 2 Find the amplitudes of electric and magnetic fields in the beam.  
    
Ans:    4.3 × 10 8 N/C 1.44 T

Q.8    The intensity of the sunlight reaching the earth is 1380 W/m2 Assume this light to be a plane monochromatic  wave Find the amplitudes of electric and magnetic fields in this wave.  
    
Ans:    1.02 × 10 3 N/C , 3.40 × 10 – 6 T 
 



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