Question

The magnetic field in a plane electromagnetic wave is \[ B_y = (3.5 \times 10^{-7}) \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{T}. \] The corresponding electric field will be:

• A. \( E_y = 1.17 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• B. \( E_z = 105 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• C. \( E_z = 1.17 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)
• D. \( E_y = 10.5 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m} \)

Answer 1

The Correct Option is B

To find the corresponding electric field for the given magnetic field of an electromagnetic wave, we need to use the relationship between the electric and magnetic fields in electromagnetic waves, given by the equation:

\(E = cB\)

where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

Given the magnetic field component:

\(B_y = (3.5 \times 10^{-7}) \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{T}\)

We can calculate the corresponding electric field as follows:

• Using the equation \(E = cB\), substitute \( c = 3 \times 10^8 \, \text{m/s} \) and the value of \( B_y \). We have:

\(E = 3 \times 10^8 \times 3.5 \times 10^{-7}\)

• Calculate the result:

\(E = 1.05 \times 10^2 = 105\)

Thus, the electric field component will be:

\(E = 105 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m}\)

Since the wave's magnetic field is in the \( y \)-direction, the electric field will be perpendicular to both the direction of wave propagation \( (x) \) and the magnetic field \( (y) \). Hence, it is in the \( z \)-direction.

Therefore, the correct answer is:

\(E_z = 105 \sin \left( 1.5 \times 10^3 x + 0.5 \times 10^{11} t \right) \, \text{V/m}\)

Answer 2

Solution: For an electromagnetic wave, the magnetic and electric fields are related by the equation:

\( E = cB \),

where \( c \) is the speed of light in a vacuum.

Given:

\( B_y = (3.5 \times 10^{-7}) \sin (1.5 \times 10^3 x + 0.5 \times 10^{11} t) \, \text{T} \),

we know that the amplitude of the electric field is:

\( E_z = c B_y = (3 \times 10^8)(3.5 \times 10^{-7}) \, \text{V/m} = 105 \, \text{V/m}. \)

Thus, the correct electric field is:

\( E_z = 105 \sin (1.5 \times 10^3 x + 0.5 \times 10^{11} t) \, \text{V/m}. \)