The electric field of a plane polarized electromagnetic wave in free space at time t=0 is given by the expression E→x, y=10j^cos6x+8z. The magnetic field B→x, z, t is given by (c is the velocity of light.)

Correct answer: (D) 1/c6k^-8i^cos6x+8z-10 ct. E=E_0cosωt-K →· r → ∴ At any time t, K→=6i^+8k^ B_0=E_0/c=10/c λ=(2π)/k=(2π)/10=(π)/5 Also, E→×B→=c→ ∴ω=2πf=(2πc)/(λ) i^ j^ k^ 0 1 0 B_x B_y B_z =(6i^+8k^)/10 ω=(2π×c)/(π5)=10c ∴i^B_z-0+j^0+k^-B_x=0.6i^-0.8k^ Comparing the coefficients on both sides, ∴B_z=0.6; B_x=-0.8 B_0→=10/c-0.8i^+0.6k^=(-8i^+6k^)/c So the magnetic field,B→=1/c-8i^+6k^cos⁡6x+8z-10ct tesla

Q81 / 90 · JEE Main 2019 (10 Jan Shift 2) · Physics

The electric field of a plane polarized electromagnetic wave in free space at time $t = 0$ is given by the expression $\vec{E} (x, y) = 10 \hat{j} c o s (6 x + 8 z)$ . The magnetic field $\vec{B} (x, z, t)$ is given by ( $c$ is the velocity of light.)

A
$\frac{1}{c} (6 \hat{k} - 8 \hat{i}) \cos (6 x + 8 z + 10 c t)$
B
$\frac{1}{c} (6 \hat{k} + 8 \hat{i}) \cos (6 x + 8 z - 10 c t)$
C
$\frac{1}{c} (6 \hat{k} + 8 \hat{i}) \cos (6 x - 8 z + 10 c t)$
Correct answer
D
$\frac{1}{c} (6 \hat{k} - 8 \hat{i}) c o s (6 x + 8 z - 10 c t)$

Solution

E = E_{0} \cos (ω t - \vec{K} \cdot \vec{r})

∴

At any time

t

\vec{K} = 6 \hat{i} + 8 \hat{k}

B_{0} = \frac{E_{0}}{c} = \frac{10}{c}

λ = \frac{2 π}{k} = \frac{2 π}{10} = \frac{π}{5}

Also,

\vec{E} \times \vec{B} = \vec{c}

∴ ω = 2 π f = \frac{2 π c}{λ}

|\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & 0 \\ B_{x} & B_{y} & B_{z} \end{matrix}| = \frac{6 \hat{i} + 8 \hat{k}}{10}

ω = \frac{2 π \times c}{(\frac{π}{5})} = 10 c

∴ \hat{i} [B_{z} - 0] + \hat{j} (0) + \hat{k} (- B_{x}) = 0.6 \hat{i} - 0.8 \hat{k}

Comparing the coefficients on both sides,

∴ B_{z} = 0.6; B_{x} = - 0.8

|\vec{B_{0}}| = \frac{10}{c} (- 0.8 \hat{i} + 0.6 \hat{k}) = \frac{- 8 \hat{i} + 6 \hat{k}}{c}

So the magnetic field,

\vec{B} = \frac{1}{c} (- 8 \hat{i} + 6 \hat{k}) \cos (6 x + 8 z - 10 c t) tesla

Hence, the correct answer is option: 'D'.

The electric field of a plane polarized electromagnetic wave in free space at time t=0 is given by the expression E→x, y=10j^cos6x+8z. The magnetic field B→x, z, t is given by (c is the velocity of light.)

Solution

Relevant Questions

The electric field of a plane polarized electromagnetic wave in free space at time $t = 0$ is given by the expression $\vec{E} (x, y) = 10 \hat{j} c o s (6 x + 8 z)$ . The magnetic field $\vec{B} (x, z, t)$ is given by ( $c$ is the velocity of light.)