If the electric field and magnetic field of an electromagnetic wave are related as `B= ( E)/( c)` where the symbols have their usual meanings and the energy in a given volume of space due to the electric field part is U, then the energy due to the magnetic field part will be

Options

A. `(U )/(c )`
B. `( U)/( c^2)`
C. `( U)/(2)`
D. U

Answer

To solve the problem, we need to find the energy due to the magnetic field part of an electromagnetic wave, given that the energy due to the electric field part is \( U \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that the magnetic field \( B \) is related to the electric field \( E \) by the equation:
\[ B = \frac{E}{c} \]
where \( c \) is the speed of light.

2. **Energy in Electric Field**:
The energy density \( U_E \) (energy per unit volume) in the electric field is given by:
\[ U_E = \frac{1}{2} \epsilon_0 E^2 \]
where \( \epsilon_0 \) is the permittivity of free space.

3. **Energy in Magnetic Field**:
The energy density \( U_B \) in the magnetic field is given by:
\[ U_B = \frac{1}{2} \frac{B^2}{\mu_0} \]
where \( \mu_0 \) is the permeability of free space.

4. **Substituting for \( B \)**:
We can substitute the expression for \( B \) into the equation for \( U_B \):
\[ U_B = \frac{1}{2} \frac{(B)^2}{\mu_0} = \frac{1}{2} \frac{\left(\frac{E}{c}\right)^2}{\mu_0} \]
This simplifies to:
\[ U_B = \frac{1}{2} \frac{E^2}{c^2 \mu_0} \]

5. **Using the Relationship between \( c \), \( \mu_0 \), and \( \epsilon_0 \)**:
We know that:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \implies c^2 = \frac{1}{\mu_0 \epsilon_0} \]
Therefore, we can replace \( c^2 \) in our expression for \( U_B \):
\[ U_B = \frac{1}{2} \frac{E^2}{\frac{1}{\mu_0 \epsilon_0}} = \frac{1}{2} \mu_0 \epsilon_0 E^2 \]

6. **Relating \( U_B \) to \( U \)**:
Since we already have the expression for \( U_E \):
\[ U_E = \frac{1}{2} \epsilon_0 E^2 \]
We can see that:
\[ U_B = \frac{1}{2} \mu_0 \epsilon_0 E^2 = U_E \]
Thus, we find that:
\[ U_B = U \]

### Final Answer:
The energy due to the magnetic field part will be equal to \( U \).

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