wo spheres of radii 3 cm and 8 cm are charged ith \( 4 \times 10^{-9} \mathrm{C} \) and \( 16 \times 10^{-9} \mathrm{C} \) of electricity spectively. If they are joined by a wire, how uch charge will flow from one to the othet

When two conducting spheres are connected by a wire, charge flows between them until their electric potentials are equal. Here's how to determine the amount of charge that flows: ### Given: - **Sphere 1:** Radius \( R_1 = 3 \) cm, Charge \( Q_1 = 4 \times 10^{-9} \) C - **Sphere 2:** Radius \( R_2 = 8 \) cm, Charge \( Q_2 = 16 \times 10^{-9} \) C ### Step-by-Step Solution: 1. **Electric Potential of a Sphere:** The electric potential \( V \) of a charged sphere is given by: \[ V = \frac{kQ}{R} \] where: - \( k \) is Coulomb's constant. - \( Q \) is the charge. - \( R \) is the radius of the sphere. 2. **Equalizing Potentials:** When the spheres are connected, their potentials become equal: \[ \frac{kQ_1'}{R_1} = \frac{kQ_2'}{R_2} \] Simplifying: \[ \frac{Q_1'}{3} = \frac{Q_2'}{8} \] \[ Q_1' = \frac{3}{8} Q_2' \] 3. **Conservation of Charge:** The total charge before and after connecting the spheres remains the same: \[ Q_1' + Q_2' = Q_1 + Q_2 = 4 \times 10^{-9} \, \text{C} + 16 \times 10^{-9} \, \text{C} = 20 \times 10^{-9} \, \text{C} \] 4. **Solving for \( Q_2' \):** Substitute \( Q_1' = \frac{3}{8} Q_2' \) into the conservation equation: \[ \frac{3}{8} Q_2' + Q_2' = 20 \times 10^{-9} \, \text{C} \] \[ \frac{11}{8} Q_2' = 20 \times 10^{-9} \, \text{C} \] \[ Q_2' = \frac{20 \times 10^{-9} \times 8}{11} \approx 14.545 \times 10^{-9} \, \text{C} \] 5. **Determining Charge Flow:** - **Sphere 2 (initially 16 × 10⁻⁹ C)** transfers charge to reach \( Q_2' \): \[ \Delta Q = 16 \times 10^{-9} \, \text{C} - 14.545 \times 10^{-9} \, \text{C} \approx 1.455 \times 10^{-9} \, \text{C} \] - **Sphere 1 (initially 4 × 10⁻⁹ C)** gains this charge: \[ Q_1' = 4 \times 10^{-9} \, \text{C} + 1.455 \times 10^{-9} \, \text{C} \approx 5.455 \times 10^{-9} \, \text{C} \] ### **Final Answer:** Approximately 1.45 × 10⁻⁹ C of charge flows from the 8 cm sphere to the 3 cm sphere.