A particle of charge `q` and velocity v passes undeflected through a space with non-zero electric field E and magnetic field B. The undeflecting conditions will hold, if

Options

A. signs of both q and E are reversed
B. signs of both q and B are reversed
C. Both B and E are changed in magnitude but keeping the product of B and E fixed
D. Both B and E are doubled in magnitude

Answer

To solve the problem of a charged particle passing undeflected through an electric field \( E \) and a magnetic field \( B \), we need to analyze the forces acting on the particle.

### Step-by-Step Solution:

1. **Understanding Forces Acting on the Particle**:
- A charged particle experiences two forces when moving through electric and magnetic fields:
- The **electric force** \( F_E \) is given by:
\[ F_E = qE \]
- The **magnetic force** \( F_B \) is given by:
\[ F_B = q(v \times B) \]
- For the particle to pass undeflected, the net force acting on it must be zero:
\[ F_E + F_B = 0 \]

2. **Condition for Undeflected Motion**:
- This implies that the magnitudes of the electric and magnetic forces must be equal and opposite:
\[ F_E = F_B \]
- Therefore, we have:
\[ qE = qvB \]

3. **Simplifying the Equation**:
- We can cancel \( q \) from both sides (assuming \( q \neq 0 \)):
\[ E = vB \]

4. **Conclusion**:
- The condition for the particle to pass undeflected through the fields is:
\[ E = vB \]
- This means that the electric field \( E \) must be equal to the product of the velocity \( v \) of the particle and the magnetic field \( B \).