A particle having charge q moves with a velocity v through a region in which both an electric field `vecE` and a magnetic field B are present. The force on the particle is

Options

A. `qvecE+q(vecBxxvecv)`
B. `qvecE.(vecBxxvecv)`
C. `qvecv+q(vecExxvecB)`
D. `qvecE+q(vecvxxvecB)`

Answer

To solve the problem of finding the total force on a charged particle moving through electric and magnetic fields, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Particle**:
- The particle experiences two types of forces: one due to the electric field and another due to the magnetic field.

2. **Calculate the Electric Force**:
- The force on a charged particle due to an electric field \( \vec{E} \) is given by the formula:
\[ \vec{F}_E = q \vec{E} \]
- Here, \( q \) is the charge of the particle and \( \vec{E} \) is the electric field vector.

3. **Calculate the Magnetic Force**:
- The force on a charged particle moving with a velocity \( \vec{v} \) in a magnetic field \( \vec{B} \) is given by the formula:
\[ \vec{F}_B = q (\vec{v} \times \vec{B}) \]
- In this equation, \( \times \) denotes the cross product, which indicates that the direction of the force is perpendicular to both the velocity and the magnetic field.

4. **Combine the Forces**:
- The total force \( \vec{F}_{\text{total}} \) acting on the particle is the vector sum of the electric and magnetic forces:
\[ \vec{F}_{\text{total}} = \vec{F}_E + \vec{F}_B \]
- Substituting the expressions for \( \vec{F}_E \) and \( \vec{F}_B \):
\[ \vec{F}_{\text{total}} = q \vec{E} + q (\vec{v} \times \vec{B}) \]

5. **Final Expression**:
- Thus, the total force on the particle can be expressed as:
\[ \vec{F}_{\text{total}} = q \vec{E} + q (\vec{v} \times \vec{B}) \]
- This expression is known as the Lorentz force.

6. **Conclusion**:
- After evaluating the options provided in the question, we conclude that the correct answer is the expression for the total force, which is:
\[ \vec{F}_{\text{total}} = q \vec{E} + q (\vec{v} \times \vec{B}) \]