Question

The electric field in a plane electromagnetic wave is given by : \( E_y = 69 \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \) V/m. The expression for magnetic field associated with this electromagnetic wave is_________ T.

• A. \( B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \)
• B. \( B_y = 69 \sin[0.6 \times 10^3 x + 1.8 \times 10^{11} t] \)
• C. \( B_z = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x + 1.8 \times 10^{11} t] \)
• D. \( B_y = 2.3 \times 10^{-7} \sin[0.6 \times 10^3 x - 1.8 \times 10^{11} t] \)

Hint:

To quickly find the direction of B, use the right-hand rule or the cyclic relation \(\hat{i} \times \hat{j} = \hat{k}\). Here, propagation is +x (\(\hat{i}\)) and E-field is +y (\(\hat{j}\)). So B must be +z (\(\hat{k}\)). For the amplitude, simply divide \(E_0\) by \(3 \times 10^8\). The sinusoidal part is always the same for both E and B.

Answer 1

The Correct Option is A

To determine the expression for the magnetic field associated with a given electromagnetic wave, we need to consider the relationship between the electric and magnetic fields in an electromagnetic wave.

The electromagnetic wave is propagating in the x-direction, as indicated by the argument of the sine function in the given electric field expression:

E_y = 69 \sin(0.6 \times 10^3 x - 1.8 \times 10^{11} t) \, \text{V/m}

In a plane electromagnetic wave, the electric field E and the magnetic field B are mutually perpendicular and also perpendicular to the direction of wave propagation. This means:

• E_y is the component of the electric field.
• B_z will be the component of the magnetic field, as the wave is propagating in the x-direction.

Using the relationship between the electric and magnetic fields in an electromagnetic wave:

c = \frac{E}{B}

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

Rearranging the equation, we find:

B = \frac{E}{c}

Substituting the given value of the electric field:

B_z = \frac{69 \, \text{V/m}}{3 \times 10^8 \, \text{m/s}} = 2.3 \times 10^{-7} \, \text{T}

Thus, the expression for the magnetic field associated with this electromagnetic wave is:

B_z = 2.3 \times 10^{-7} \sin(0.6 \times 10^3 x - 1.8 \times 10^{11} t) \, \text{T}

The correct answer is:

B_z = 2.3 \times 10^{-7} \sin(0.6 \times 10^3 x - 1.8 \times 10^{11} t)

This matches the options given, concluding that this option is the correct expression for the magnetic field.