A certain region has an electric field `vecE = (2hati-3hatj)` N/C and a uniform magnetic field `B = (5 hati+3hatj+4k)` . The force experience by a charge IC moving with velocity `(i+2hatj)`

Options

A. `(10 hati-7hatj-7hatk)`
B. `(10 hati+ 7hatj+7hatk)`
C. `(-10 hati+7hatj+7hatk)`
D. `(10 hati+7hatj-7hatk)`

Explanation

To solve the problem of finding the force experienced by a charge moving in an electric and magnetic field, we will use the Lorentz force equation:

\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]

Where:
- \(\vec{F}\) is the total force on the charge,
- \(q\) is the charge,
- \(\vec{E}\) is the electric field,
- \(\vec{v}\) is the velocity of the charge,
- \(\vec{B}\) is the magnetic field.

Given:
- \(\vec{E} = 2\hat{i} - 3\hat{j}\) N/C,
- \(\vec{B} = 5\hat{i} + 3\hat{j} + 4\hat{k}\) T,
- \(\vec{v} = \hat{i} + 2\hat{j}\) m/s,
- \(q = 1\) C.

### Step 1: Calculate the cross product \(\vec{v} \times \vec{B}\)

To find \(\vec{v} \times \vec{B}\), we set up the determinant:

\[ \vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 0 \\ 5 & 3 & 4 \end{vmatrix} \]

Calculating this determinant:

\[ \vec{v} \times \vec{B} = \hat{i} \begin{vmatrix} 2 & 0 \\ 3 & 4 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 0 \\ 5 & 4 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 5 & 3 \end{vmatrix} \]

Calculating the 2x2 determinants:

1. \(\begin{vmatrix} 2 & 0 \\ 3 & 4 \end{vmatrix} = (2)(4) - (0)(3) = 8\)
2. \(\begin{vmatrix} 1 & 0 \\ 5 & 4 \end{vmatrix} = (1)(4) - (0)(5) = 4\)
3. \(\begin{vmatrix} 1 & 2 \\ 5 & 3 \end{vmatrix} = (1)(3) - (2)(5) = 3 - 10 = -7\)

Now substituting back:

\[ \vec{v} \times \vec{B} = 8\hat{i} - 4\hat{j} - 7\hat{k} \]

### Step 2: Calculate the total force \(\vec{F}\)

Now we substitute \(\vec{E}\) and \(\vec{v} \times \vec{B}\) into the Lorentz force equation:

\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]

Substituting \(q = 1\):

\[ \vec{F} = 1 \left( (2\hat{i} - 3\hat{j}) + (8\hat{i} - 4\hat{j} - 7\hat{k}) \right) \]

Combine the vectors:

\[ \vec{F} = (2 + 8)\hat{i} + (-3 - 4)\hat{j} - 7\hat{k} \]

\[ \vec{F} = 10\hat{i} - 7\hat{j} - 7\hat{k} \]

### Final Result

The force experienced by the charge is:

\[ \vec{F} = 10\hat{i} - 7\hat{j} - 7\hat{k} \text{ N} \]