A charged particle of mass `5 xx 10^(-5) kg` is held stationary in space by placing it in an electric field of strength `10^(7) NC^(-1)` directed vertically downwards. The charge on the particle is

Options

A. `-20 xx 10^(-5) muC`
B. `-5 xx 10^(-5) muC`
C. `5 xx 10^(-5) muC`
D. `20 xx 10^(-5) muC`

Answer

To solve the problem, we need to find the charge on a particle that is held stationary in an electric field. The steps to find the solution are as follows:

### Step 1: Understand the forces acting on the charged particle
The charged particle experiences two forces:
1. The gravitational force (weight) acting downwards, given by \( F_g = mg \).
2. The electric force acting upwards, given by \( F_e = QE \).

### Step 2: Set up the equilibrium condition
Since the particle is stationary, the upward electric force must balance the downward gravitational force:
\[ F_e = F_g \]
This can be expressed as:
\[ QE = mg \]

### Step 3: Substitute known values
We know:
- Mass \( m = 5 \times 10^{-5} \, \text{kg} \)
- Acceleration due to gravity \( g \approx 10 \, \text{m/s}^2 \)
- Electric field strength \( E = 10^7 \, \text{N/C} \)

Substituting these values into the equation gives:
\[ Q \cdot 10^7 = (5 \times 10^{-5}) \cdot 10 \]

### Step 4: Calculate the gravitational force
Calculating the gravitational force:
\[ F_g = mg = (5 \times 10^{-5} \, \text{kg}) \cdot (10 \, \text{m/s}^2) = 5 \times 10^{-4} \, \text{N} \]

### Step 5: Solve for the charge \( Q \)
Now we can rearrange the equilibrium equation to solve for \( Q \):
\[ Q = \frac{F_g}{E} = \frac{5 \times 10^{-4}}{10^7} \]
\[ Q = 5 \times 10^{-4} \times 10^{-7} = 5 \times 10^{-11} \, \text{C} \]

### Step 6: Convert to microcoulombs
Since the problem asks for the charge in microcoulombs, we convert:
\[ 1 \, \text{C} = 10^6 \, \mu\text{C} \]
Thus,
\[ Q = 5 \times 10^{-11} \, \text{C} = 5 \times 10^{-5} \, \mu\text{C} \]

### Step 7: Determine the sign of the charge
Since the electric force must act upwards to balance the weight (which acts downwards), the charge must be negative (as the electric field is directed downwards).

### Final Answer
The charge on the particle is:
\[ Q = -5 \times 10^{-5} \, \mu\text{C} \]

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