50.0pJ of energy is stored in a 3.00cm × 3.00cm × 3.00cm region of uniform electric field.
To determine the electric field strength in this region, we can use the formula for energy stored in an electric field:
Energy = (1/2) × ε₀ × E² × V,
Where:
- Energy is the amount of energy stored in the electric field (given as 50.0 pJ),
- ε₀ is the permittivity of free space (a constant value of 8.85 x 10⁻¹² C²/N·m²),
- E is the electric field strength (what we want to find), and
- V is the volume of the region in the electric field (given as 3.00 cm × 3.00 cm × 3.00 cm).
First, let's convert the volume to meters (m):
Volume = (3.00 cm) × (3.00 cm) × (3.00 cm)
= 27.00 cm³,
Since there are 100 cm in 1 m:
Volume = (27.00 cm³) × (1 m / 100 cm)³
= 0.00027 m³.
Now, let's rearrange the formula to solve for E:
E² = (2 × Energy) / (ε₀ × V).
Substituting the given values:
E² = (2 × 50.0 pJ) / (8.85 × 10⁻¹² C²/N·m² × 0.00027 m³).
Calculating this equation:
E² ≈ 3.79 × 10²² N²/C².
To find E, we take the square root of both sides:
E = √(3.79 × 10²² N²/C²).
E ≈ 1.95 × 10¹¹ N/C.
Therefore, the electric field strength in that region is approximately 1.95 × 10¹¹ N/C.