Find the capacitnace of a conducting sphere of radius 10 cm situated in air. How much charge is required to raise it to a potential 1000 volt?

Answer

To solve the problem step by step, we will follow the outlined process to find the capacitance of a conducting sphere and the charge required to raise it to a certain potential.

### Step 1: Identify the given values
- Radius of the sphere, \( R = 10 \, \text{cm} = 10 \times 10^{-2} \, \text{m} = 0.1 \, \text{m} \)
- Potential, \( V = 1000 \, \text{V} \)

### Step 2: Use the formula for the capacitance of a conducting sphere
The capacitance \( C \) of a conducting sphere is given by the formula:
\[ C = 4 \pi \epsilon_0 R \]
where \( \epsilon_0 \) (the permittivity of free space) is approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

### Step 3: Substitute the values into the formula
Substituting the values into the capacitance formula:
\[ C = 4 \pi (8.85 \times 10^{-12} \, \text{F/m}) (0.1 \, \text{m}) \]

### Step 4: Calculate the capacitance
Calculating the capacitance:
\[ C = 4 \times 3.14 \times (8.85 \times 10^{-12}) \times (0.1) \]
\[ C \approx 1.112 \times 10^{-12} \, \text{F} \]
To express this in picoFarads (pF):
\[ C \approx 11.12 \, \text{pF} \]

### Step 5: Calculate the charge required to raise the potential
The charge \( Q \) required to raise the potential is given by the formula:
\[ Q = C \times V \]
Substituting the values:
\[ Q = (1.112 \times 10^{-12} \, \text{F}) \times (1000 \, \text{V}) \]

### Step 6: Calculate the charge
Calculating the charge:
\[ Q = 1.112 \times 10^{-9} \, \text{C} \]
To express this in nanoCoulombs (nC):
\[ Q \approx 11.12 \, \text{nC} \]

### Final Answers
- Capacitance \( C \approx 11.12 \, \text{pF} \)
- Charge \( Q \approx 11.12 \, \text{nC} \)