The energy stored in the electric field produced by a metal sphere is 4.5 J. If the sphere contains `4mu C` charge, its radius will be:

Options

A. 32 mm
B. 16 mm
C. 28 mm
D. 20 mm

Answer

To find the radius of a metal sphere given the energy stored in its electric field and the charge it contains, we can follow these steps:

### Step 1: Write down the given values
- Energy stored in the electric field, \( E = 4.5 \, \text{J} \)
- Charge contained by the sphere, \( Q = 4 \, \mu C = 4 \times 10^{-6} \, C \)

### Step 2: Use the formula for energy stored in a capacitor
The energy stored in a capacitor can be expressed as:
\[ E = \frac{Q^2}{2C} \]
where \( C \) is the capacitance of the sphere.

### Step 3: Write the formula for the capacitance of a sphere
The capacitance \( C \) of a sphere of radius \( r \) is given by:
\[ C = 4 \pi \epsilon_0 r \]
where \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

### Step 4: Substitute the capacitance into the energy formula
Substituting \( C \) into the energy formula gives:
\[ E = \frac{Q^2}{2(4 \pi \epsilon_0 r)} \]

### Step 5: Rearrange the equation to solve for \( r \)
Rearranging the equation to solve for \( r \):
\[ r = \frac{Q^2}{8 \pi \epsilon_0 E} \]

### Step 6: Substitute the known values
Now, substituting the known values into the equation:
\[ r = \frac{(4 \times 10^{-6})^2}{8 \pi (8.85 \times 10^{-12}) (4.5)} \]

### Step 7: Calculate \( r \)
Calculating the numerator:
\[ (4 \times 10^{-6})^2 = 16 \times 10^{-12} \, C^2 \]
Calculating the denominator:
\[ 8 \pi (8.85 \times 10^{-12}) (4.5) \approx 1.000 \times 10^{-10} \, F \]
Now substituting these values:
\[ r = \frac{16 \times 10^{-12}}{1.000 \times 10^{-10}} = 0.16 \, m \]

### Step 8: Convert to millimeters
To convert meters to millimeters, multiply by \( 1000 \):
\[ r = 0.16 \, m \times 1000 = 160 \, mm \]

### Final Answer
The radius of the sphere is:
\[ \boxed{16 \, mm} \]