A uniform magnetic existing in space is `vec B = (2 hat I + 3 hat j - 5 hat k)` tesla . At one instant of time, the force experienced by a `2 mu C` point charge is `vec F = (x hat I - 2 hat j+ 2 hat k)` N. What should be the value of x ?

Options

A. 4
B. 8
C. 16
D. 0

Explanation

To solve the problem, we need to find the value of \( x \) in the force vector \( \vec{F} = (x \hat{i} - 2 \hat{j} + 2 \hat{k}) \) N, given the magnetic field \( \vec{B} = (2 \hat{i} + 3 \hat{j} - 5 \hat{k}) \) T and the charge \( q = 2 \, \mu C \).

### Step-by-Step Solution:

1. **Understand the Force on a Charge in a Magnetic Field**:
The force experienced by a charged particle moving in a magnetic field is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
where \( \vec{F} \) is the force, \( q \) is the charge, \( \vec{v} \) is the velocity of the charge, and \( \vec{B} \) is the magnetic field.

2. **Use the Perpendicular Condition**:
Since the force \( \vec{F} \) is perpendicular to the magnetic field \( \vec{B} \), we can use the dot product:
\[ \vec{F} \cdot \vec{B} = 0 \]
This means that the dot product of the force vector and the magnetic field vector must equal zero.

3. **Calculate the Dot Product**:
Substitute \( \vec{F} \) and \( \vec{B} \) into the dot product:
\[ (x \hat{i} - 2 \hat{j} + 2 \hat{k}) \cdot (2 \hat{i} + 3 \hat{j} - 5 \hat{k}) = 0 \]
Calculate the dot product:
\[ = x \cdot 2 + (-2) \cdot 3 + 2 \cdot (-5) \]
Simplifying this gives:
\[ = 2x - 6 - 10 = 2x - 16 \]

4. **Set the Dot Product to Zero**:
Set the equation equal to zero:
\[ 2x - 16 = 0 \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ 2x = 16 \]
Dividing both sides by 2 yields:
\[ x = 8 \]

### Final Answer:
Thus, the value of \( x \) is \( 8 \).

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