To find the magnitude of the source charge that generates a given electric field, we can use Coulomb's law. The electric field \( E \) created by a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ E = \frac{k \cdot |Q|}{r^2} \]
where:
- \( E \) is the electric field (in N/C),
- \( k \) is Coulomb's constant (\( k \approx 8.99 \times 10^9 , \text{N m}^2/\text{C}^2 \)),
- \( Q \) is the source charge (in coulombs), and
- \( r \) is the distance from the charge (in meters).
Given:
- \( E = 1236 , \text{N/C} \)
- \( r = 4 , \text{m} \)
We can rearrange the formula to solve for \( Q \):
\[ |Q| = \frac{E \cdot r^2}{k} \]
Substituting the known values:
\[ |Q| = \frac{1236 , \text{N/C} \cdot (4 , \text{m})^2}{8.99 \times 10^9 , \text{N m}^2/\text{C}^2} \]
Calculating \( (4 , \text{m})^2 \):
\[ (4 , \text{m})^2 = 16 , \text{m}^2 \]
Now substitute this back into the equation:
\[ |Q| = \frac{1236 , \text{N/C} \cdot 16 , \text{m}^2}{8.99 \times 10^9 , \text{N m}^2/\text{C}^2} \]
Calculating the numerator:
\[ 1236 \cdot 16 = 19776 \]
Now substituting this into the equation for \( Q \):
\[ |Q| = \frac{19776}{8.99 \times 10^9} \]
Calculating the value:
\[ |Q| \approx \frac{19776}{8.99 \times 10^9} \approx 2.20 \times 10^{-6} , \text{C} \]
Now, converting this into microcoulombs (\( \mu C \)):
\[ |Q| \approx 2.20 , \mu C \]
Therefore, the magnitude of the source charge is:
\[ \text{2.2 µC} \]
So, the correct answer is 2.2 µC.