What is the charge on the bead?

Steps to Solve

1. Identify the Given Values The bead has a mass of ( m = 1.0 , \text{g} = 0.001 , \text{kg} ), an electric field strength of ( E = 200,000 , \text{N/C} ), and it accelerates upwards with ( a = 20 , \text{m/s}^2 ).

2. Calculate the Weight of the Bead The weight ( W ) of the bead, which acts downward, can be calculated using the equation: $$ W = mg $$ where ( g ) is the acceleration due to gravity, approximately ( 9.81 , \text{m/s}^2 ). Substituting the values in: $$ W = 0.001 , \text{kg} \times 9.81 , \text{m/s}^2 = 0.00981 , \text{N} $$

3. Calculate the Net Force Acting on the Bead Using Newton's second law ( F = ma ) to find the net force ( F_{\text{net}} ) acting upwards: $$ F_{\text{net}} = ma = 0.001 , \text{kg} \times 20 , \text{m/s}^2 = 0.02 , \text{N} $$

4. Set Up the Equation of Forces The upward net force must overcome the downward weight of the bead plus the upward electric force ( F_e ). Thus, $$ F_{\text{net}} = F_e - W $$ Rearranging gives: $$ F_e = F_{\text{net}} + W $$

5. Substituting for the Electric Force The electric force ( F_e ) is given by: $$ F_e = qE $$ where ( q ) is the charge on the bead.

6. Combine the Equations Substituting ( F_e ): $$ qE = F_{\text{net}} + W $$ Substituting the previous equations we have: $$ qE = 0.02 , \text{N} + 0.00981 , \text{N} $$

7. Solve for Charge ( q ) Summing the forces: $$ q \cdot 200,000 , \text{N/C} = 0.02981 , \text{N} $$ Dividing by ( E ): $$ q = \frac{0.02981 , \text{N}}{200,000 , \text{N/C}} $$

8. Final Calculation Calculating ( q ): $$ q \approx 1.4905 \times 10^{-7} , \text{C} $$

9. Convert to Nanocoulombs Converting to nanocoulombs: $$ q \approx 149.05 , \text{nC} $$

Since the bead accelerates upwards in an upward electric field, it indicates a negative charge. Thus, rounding gives: $$ q \approx -149 , \text{nC} $$