Electric field in space is given by `vec(E(t)) = E_0 (i+j)/sqrt2 cos(omegat+Kz)`. A positively charged particle at `(0, 0, pi/K)` is given velocity `v_0 hatk` at t = 0. Direction of force acting on particle is

Options

A. f = 0
B. antiparallel to `(hati+hatj)/sqrt2`
C. parallel to `(hati+hatj)/sqrt2`
D. `hatk`

Explanation

To solve the problem, we need to determine the direction of the force acting on a positively charged particle in an electric field given by:

\[ \vec{E}(t) = E_0 \frac{(i + j)}{\sqrt{2}} \cos(\omega t + kz) \]

The particle is initially located at \((0, 0, \frac{\pi}{k})\) and has an initial velocity of \(\vec{v} = v_0 \hat{k}\) at \(t = 0\).

### Step 1: Calculate the Electric Field at \(t = 0\)

First, we need to find the electric field at the position of the particle when \(t = 0\):

\[ \vec{E}(0) = E_0 \frac{(i + j)}{\sqrt{2}} \cos(0 + k \cdot \frac{\pi}{k}) \]

This simplifies to:

\[ \vec{E}(0) = E_0 \frac{(i + j)}{\sqrt{2}} \cos(\pi) \]

Since \(\cos(\pi) = -1\), we have:

\[ \vec{E}(0) = -E_0 \frac{(i + j)}{\sqrt{2}} \]

### Step 2: Determine the Direction of the Electric Field

The direction of the electric field \(\vec{E}(0)\) is:

\[ \vec{E}(0) = -\frac{E_0}{\sqrt{2}} (i + j) \]

This indicates that the electric field is directed in the opposite direction of the vector \((i + j)\).

### Step 3: Calculate the Magnetic Field Direction

The electromagnetic wave propagates in the \(-z\) direction (as indicated by the \(\cos(\omega t + kz)\) term). The direction of the magnetic field \(\vec{B}\) can be determined using the right-hand rule, where \(\vec{E}\) and \(\vec{B}\) are perpendicular to the direction of wave propagation.

Using the relation:

\[ \vec{E} \times \vec{B} \propto \hat{k} \]

We can find the direction of \(\vec{B}\):

1. The electric field \(\vec{E}\) points in the direction of \(-\frac{E_0}{\sqrt{2}} (i + j)\).
2. The wave propagates in the \(-z\) direction, so we can use the right-hand rule to find \(\vec{B}\).

### Step 4: Calculate the Magnetic Force on the Particle

The force acting on a charged particle in an electric field is given by:

\[ \vec{F}_E = q \vec{E} \]

The magnetic force is given by:

\[ \vec{F}_B = q \vec{v} \times \vec{B} \]

### Step 5: Determine the Total Force Direction

The total force on the particle is:

\[ \vec{F} = \vec{F}_E + \vec{F}_B \]

1. Calculate \(\vec{F}_E\) using the electric field we found.
2. Calculate \(\vec{F}_B\) using the velocity \(\vec{v} = v_0 \hat{k}\) and the direction of \(\vec{B}\) we determined.

### Conclusion: Direction of the Force

The direction of the total force \(\vec{F}\) will be the vector sum of \(\vec{F}_E\) and \(\vec{F}_B\). Since both forces are vectors, we can find their resultant direction.