At a certain distance from a point charge, the field intensity is `500 V//m` and the potential is `-3000 V`. The distance to the charge and the magnitude of the charge respectively are

Options

A. 6 m
B. 12 m
C. 36 m
D. 144 m

Answer

To solve the problem, we need to find the distance from the point charge and the magnitude of the charge using the given electric field intensity and potential.

### Step-by-step Solution:

1. **Understand the formulas**:
- The electric field intensity (E) due to a point charge is given by:
\[ E = \frac{kQ}{R^2} \]
- The electric potential (V) due to a point charge is given by:
\[ V = \frac{kQ}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the distance from the charge.

2. **Given values**:
- Electric field intensity, \( E = 500 \, \text{V/m} \)
- Electric potential, \( V = -3000 \, \text{V} \)

3. **Set up the equations**:
- From the potential equation:
\[ V = \frac{kQ}{R} \implies kQ = VR \]
- From the electric field equation:
\[ E = \frac{kQ}{R^2} \implies kQ = ER^2 \]

4. **Equate the two expressions for \( kQ \)**:
\[ VR = ER^2 \]

5. **Rearranging the equation**:
\[ R = \frac{V}{E} \]

6. **Substituting the known values**:
\[ R = \frac{-3000 \, \text{V}}{500 \, \text{V/m}} = -6 \, \text{m} \]
Since distance cannot be negative, we take the absolute value:
\[ R = 6 \, \text{m} \]

7. **Now, find the magnitude of the charge \( Q \)**:
- Substitute \( R \) back into one of the original equations. Using the potential equation:
\[ V = \frac{kQ}{R} \implies kQ = VR \]
- Substitute \( V \) and \( R \):
\[ kQ = (-3000 \, \text{V})(6 \, \text{m}) = -18000 \, \text{V m} \]

8. **Calculate \( Q \)**:
- Using \( k \approx 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \):
\[ Q = \frac{-18000}{k} = \frac{-18000}{8.99 \times 10^9} \approx -2.00 \times 10^{-6} \, \text{C} \]
- Therefore, the magnitude of the charge is:
\[ |Q| \approx 2.00 \, \mu\text{C} \]

### Final Answers:
- Distance to the charge \( R \): \( 6 \, \text{m} \)
- Magnitude of the charge \( |Q| \): \( 2.00 \, \mu\text{C} \)