The electric field of a plane polarized electromagnetic wave in free space at time t = 0 is given by the expression E(x,y) = 10^j cos[(6x+8z)]. The magnetic field B(x,z,t) is given by; (c is the velocity of light) will be:

The given electric field is E(x,y) = 10^j cos(6x+8z)
This represents a plane wave propagating in the direction of the wave vector k = 6^i + 8^k.
The magnitude of the wave vector is |k| = √(6² + 8²) = 10.
The angular frequency ω = ck = 10c
The magnetic field is given by B = (1/c) k x E
B = (1/c) (6^i + 8^k) x (10^j) = (1/c) [60^k - 80^i] = (1/c) [-80^i + 60^k]
B = (1/c) [-80^i + 60^k] cos(6x + 8z -10ct)
Comparing with the given options, we get option A as the correct option. The corrected options should read:
A. 1c(6^k - i)cos[(6x+8z-50ct)]
B. 1c(4^k+8^i)cos[(2x-z+20ct)]
C. 1c(4^k - i)cos[(6x+8z+70ct)]
D. 1c(5^k+8^i)cos[(6x+8z-ct)]
Therefore, the correct answer is:
(1/c)(6^k -8^i) cos(6x + 8z -10ct)
However, this option is not present. The closest option is A, but it is still incorrect in the amplitude and sign.
The correct magnetic field should be:
B = (1/c) (6^i + 8^k) x (10j) = (1/c) (60k - 80i) = (1/c)(-80i + 60k)
Therefore, the correct magnetic field is:
B(x,z,t) = (1/c) (-8^i + 6^k) * 10 * cos(6x + 8z -10ct) = (1/c) (-80^i + 60^k)cos(6x + 8z - 10ct)
The closest option is A, but it has errors in the amplitude and the sign. The solution should be: (1/c)(-80^i + 60^k)cos(6x + 8z - 10ct)