Find the energy contained in a cyllinder of cross - section `10cm^(2)` and length 50 along x - axis, if `E=50sin omega(t-x//C)` be the electric field in an electromagnetic wave

Options

A. `5.5xx10^(-12)J`
B. `1.5xx10^(-11)J`
C. `6.2xx10^(-10)J`
D. `1.1xx10^(-15)J`

Answer

To find the energy contained in a cylinder of cross-section \(10 \, \text{cm}^2\) and length \(50 \, \text{cm}\) along the x-axis, given the electric field \(E = 50 \sin(\omega(t - \frac{x}{c}))\), we can follow these steps:

### Step 1: Convert Units
Convert the cross-sectional area and length from centimeters to meters.

- Cross-sectional area:
\[ A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 1 \times 10^{-4} \, \text{m}^2 \]

- Length:
\[ L = 50 \, \text{cm} = 50 \times 10^{-2} \, \text{m} = 0.5 \, \text{m} \]

### Step 2: Calculate the Volume of the Cylinder
The volume \(V\) of the cylinder can be calculated using the formula:
\[ V = A \times L \]
Substituting the values:
\[ V = (1 \times 10^{-4} \, \text{m}^2) \times (0.5 \, \text{m}) = 5 \times 10^{-5} \, \text{m}^3 \]

### Step 3: Calculate the Energy Density
The energy density \(u\) of the electric field in an electromagnetic wave is given by:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
Where:
- \(\epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m}\) (permittivity of free space)
- \(E = 50 \, \text{V/m}\)

Substituting the values:
\[ u = \frac{1}{2} \times (8.85 \times 10^{-12}) \times (50)^2 \]
Calculating \(E^2\):
\[ E^2 = 50^2 = 2500 \]
Now substituting back:
\[ u = \frac{1}{2} \times (8.85 \times 10^{-12}) \times 2500 \]
\[ u = 1.10625 \times 10^{-8} \, \text{J/m}^3 \]

### Step 4: Calculate the Total Energy
The total energy \(U\) contained in the cylinder is given by:
\[ U = u \times V \]
Substituting the values:
\[ U = (1.10625 \times 10^{-8} \, \text{J/m}^3) \times (5 \times 10^{-5} \, \text{m}^3) \]
\[ U = 5.53125 \times 10^{-13} \, \text{J} \]

### Final Answer
The energy contained in the cylinder is approximately:
\[ U \approx 5.53 \times 10^{-13} \, \text{J} \]

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