A charged particle of mass 0.003 g is held stationary in space by placing it in a downward direction of electric field of `6xx10^(4)NC^(-1)`. Then, the magnitude of charge is

Options

A. `5xx10^(-4)C`
B. `5xx10^(-10)C`
C. `5xx10^(-6)C`
D. `5xx10^(-9)C`

Explanation

To solve the problem step by step, we will use the relationship between the forces acting on the charged particle and the electric field.

### Step 1: Understand the forces acting on the charged particle.
The charged particle is held stationary in an electric field, which means the net force acting on it is zero. The two forces acting on the particle are:
1. The gravitational force (downward).
2. The electric force (upward due to the electric field).

### Step 2: Write the equations for the forces.
The gravitational force \( F_g \) acting on the particle can be expressed as:
\[ F_g = mg \]
where:
- \( m \) = mass of the particle in kg,
- \( g \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).

The electric force \( F_e \) acting on the charged particle in the electric field \( E \) is given by:
\[ F_e = QE \]
where:
- \( Q \) = charge of the particle,
- \( E \) = electric field strength.

### Step 3: Set the forces equal to each other.
Since the particle is stationary, the upward electric force must balance the downward gravitational force:
\[ QE = mg \]

### Step 4: Solve for the charge \( Q \).
Rearranging the equation gives:
\[ Q = \frac{mg}{E} \]

### Step 5: Substitute the known values.
1. Convert the mass from grams to kilograms:
\[ m = 0.003 \, \text{g} = 0.003 \times 10^{-3} \, \text{kg} = 3 \times 10^{-6} \, \text{kg} \]
2. Use the given electric field:
\[ E = 6 \times 10^4 \, \text{N/C} \]
3. Substitute the values into the equation:
\[ Q = \frac{(3 \times 10^{-6} \, \text{kg})(9.8 \, \text{m/s}^2)}{6 \times 10^4 \, \text{N/C}} \]

### Step 6: Calculate \( Q \).
\[ Q = \frac{(3 \times 10^{-6})(9.8)}{6 \times 10^4} \]
\[ Q = \frac{29.4 \times 10^{-6}}{6 \times 10^4} \]
\[ Q = \frac{29.4}{6} \times 10^{-10} \]
\[ Q = 4.9 \times 10^{-10} \, \text{C} \]

### Final Answer:
The magnitude of the charge \( Q \) is approximately \( 5 \times 10^{-10} \, \text{C} \).

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