Direction of Propagation of Electromagnetic Wave

The direction of propagation of an electromagnetic wave is perpendicular to both the electric field and the magnetic field vectors. To determine the unit vector along the direction of propagation, we can use the right-hand rule.

Right-Hand Rule
The right-hand rule states that if the thumb of the right hand points in the direction of the electric field vector and the fingers curl in the direction of the magnetic field vector, then the palm represents the direction of propagation of the wave.

Given Information
Electric field vector, E = k
Magnetic field vector, B = 2i - 2j

Calculating the Unit Vector along the Direction of Propagation
To calculate the unit vector along the direction of propagation, we need to find the cross product of the electric field vector and the magnetic field vector.

E x B = (k x (2i - 2j))

To find the cross product, we can use the determinant method:
E x B = (k x (2i - 2j))
= [(k x 2i) - (k x 2j)]

Using the properties of cross product, we get:
E x B = (2k x i) - (2k x j)

To determine the cross product of unit vectors, we can use the right-hand rule again:
- The cross product of k and i gives the direction of j.
- The cross product of k and j gives the direction of -i.

Therefore, we have:
E x B = 2j - (-2i)
= 2j + 2i

Unit Vector along the Direction of Propagation
Since the cross product of the electric field vector and the magnetic field vector is 2j + 2i, the unit vector along the direction of propagation is the normalized form of this vector.

To normalize the vector, we divide it by its magnitude:
Unit vector along the direction of propagation = (2j + 2i) / √(2^2 + 2^2)
= (2j + 2i) / √8
= (1/√2) * (j + i)

Therefore, the unit vector along the direction of propagation of the electromagnetic wave is (1/√2) * (j + i).