Short Answer 
To calculate the electric field, first find the distance to the charge using the Pythagorean theorem, resulting in r = 11.18 cm. Next, compute the horizontal and vertical electric field components, leading to Ex = 12,731.5 N/C and Ey = 967.22 N/C, and finally determine the resultant electric field’s magnitude as 12,768.2 N/C with a direction of ≈í‚àè = 4.3¬¨‚àû.
Step 1: Calculate the Distance to the Charge
To determine the magnitude of the electric field, first calculate the distance from the charge to the point of interest. Use the Pythagorean theorem with the following steps:
- Identify the vertical and horizontal distances, which in this case are 5 cm and 10 cm respectively.
- Apply the formula for distance: r = ‚ao[(5 cm)¬≤ + (10 cm)¬≤].
- Calculate to find r = 11.18 cm or 0.1118 m.
Step 2: Calculate the Electric Field Components
Next, compute the electric field components acting in both the horizontal and vertical directions. Follow these procedures for accurate calculation:
- For the horizontal component, use the formula: Ex = [(9 x 10¬π x 3 x 10‚Aª¬π) / (0.1118)¬≤] cos(26.6) + (9 x 10¬π x 3 x 10‚Aª¬π) / (0.05)¬≤.
- Calculate to find the horizontal electric field, Ex = 12,731.5 N/C.
- For the vertical component, use Ey = [(9 x 10¬π x 3 x 10‚Aª¬π) / (0.1118)¬≤] sin(26.6).
- This results in a vertical electric field of Ey = 967.22 N/C.
Step 3: Determine the Magnitude and Direction of the Electric Field
The final step is to calculate the resultant electric field’s magnitude and direction. Perform the following calculations:
- Calculate the magnitude using E = ‚ao(Ex¬≤ + Ey¬≤), leading to E = 12,768.2 N/C.
- Determine the direction by computing tan(θ) = Ey/Ex, leading to tan(θ) = (967.22) / (12,731.5).
- Finally, compute the angle θ = arc tan(0.076) to get θ = 4.3°.