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An electromagnetic wave is propagating i...

An electromagnetic wave is propagating in vacuum along z-axis, the electric field component is given by `E_(x)=E_(0)sin (kz-omegat)`. Then magnetic components is

A

`B_(x)=(E_(0))/(C)sin (kz-omegat)`

B

`B_(y)=(B_(0))/(C)sin (kz-omegat)`

C

`B_(y)=(E_(0))/(C)sin (kz-omegat)`

D

`B_(y)=B_(0)Csin (kz-omegat)`

Text Solution

AI Generated Solution

The correct Answer is:
To find the magnetic field component of an electromagnetic wave propagating in vacuum along the z-axis, we can follow these steps: ### Step 1: Identify the given electric field The electric field component is given as: \[ E_x = E_0 \sin(kz - \omega t) \] ### Step 2: Understand the relationship between electric and magnetic fields in electromagnetic waves In a vacuum, the electric field \( \vec{E} \) and magnetic field \( \vec{B} \) are related by the speed of light \( c \): \[ c = \frac{E_0}{B_0} \] where \( E_0 \) is the amplitude of the electric field and \( B_0 \) is the amplitude of the magnetic field. ### Step 3: Calculate the amplitude of the magnetic field From the relationship above, we can express the amplitude of the magnetic field as: \[ B_0 = \frac{E_0}{c} \] ### Step 4: Determine the direction of the magnetic field Since the wave is propagating along the z-axis and the electric field is in the x-direction, we can use the right-hand rule to find the direction of the magnetic field. According to the right-hand rule: - Point your thumb in the direction of wave propagation (z-axis). - Your fingers point in the direction of the electric field (x-axis). - The palm will face in the direction of the magnetic field (y-axis). Thus, the magnetic field \( \vec{B} \) will be in the y-direction. ### Step 5: Write the equation for the magnetic field Since the magnetic field is in the y-direction and has the same sinusoidal form as the electric field, we can write: \[ B_y = B_0 \sin(kz - \omega t) \] Substituting \( B_0 \) from Step 3: \[ B_y = \frac{E_0}{c} \sin(kz - \omega t) \] ### Final Answer The magnetic field component is: \[ B_y = \frac{E_0}{c} \sin(kz - \omega t) \]
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