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

Question

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

UpStudy AI · Accepted Answer

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.
Approximately 1.45 × 10⁻⁹ C of charge flows from the 8 cm sphere to the 3 cm sphere.

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

Solución de tutoría real

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Physics Questions