A +7.5 nC point charge and a -2.0 nC point charge are 3.0 cm apart. What is the electric field strength at the midpoint between the two charges?

Steps to Solve

1. Identify the midpoint The charges are separated by a distance of 3.0 cm. The midpoint is at a distance of 1.5 cm from each charge.

2. Convert units Convert the distances from centimeters to meters: $$ 1.5 , \text{cm} = 0.015 , \text{m} $$

3. Use the formula for electric field strength The electric field strength $E$ generated by a point charge is given by the formula: $$ E = \frac{k |q|}{r^2} $$ where

• $k = 8.99 \times 10^9 , \text{N m}^2/\text{C}^2$ (Coulomb's constant),
• $q$ is the charge, and
• $r$ is the distance from the charge to the point where the field is being calculated.

4. Calculate electric field due to the +7.5 nC charge For the positive charge: $$ q_1 = 7.5 \times 10^{-9} , \text{C} $$ $$ r_1 = 0.015 , \text{m} $$ Calculate: $$ E_1 = \frac{(8.99 \times 10^9) (7.5 \times 10^{-9})}{(0.015)^2} $$

5. Calculate electric field due to the -2.0 nC charge For the negative charge: $$ q_2 = -2.0 \times 10^{-9} , \text{C} $$ $$ r_2 = 0.015 , \text{m} $$ Calculate: $$ E_2 = \frac{(8.99 \times 10^9) (2.0 \times 10^{-9})}{(0.015)^2} $$

6. Determine the direction of electric fields The electric field due to the positive charge ($E_1$) points away from it, while the electric field due to the negative charge ($E_2$) points towards it. Since they are in the same direction at the midpoint, the strengths add.

7. Calculate net electric field The total electric field strength at the midpoint is given by: $$ E_{net} = E_1 + E_2 $$

8. Perform the calculations Calculate the values from steps 4 and 5, then sum them.