The energy density in magnetic field B is proportional to

Options

A. `1/B`
B. `1/(B^(2))`
C. B
D. `B^(2)`

Answer

To solve the question regarding the energy density in a magnetic field \( B \), we will derive the relationship step by step.

### Step 1: Understand the concept of energy density
The energy density \( U \) in an electromagnetic field is the energy stored per unit volume. For electric and magnetic fields, the energy density can be expressed as:
- For electric field:
\[ U_E = \frac{1}{2} \epsilon_0 E_{\text{rms}}^2 \]
- For magnetic field:
\[ U_B = \frac{1}{2} \frac{B_{\text{rms}}^2}{\mu_0} \]

### Step 2: Relate electric and magnetic fields in electromagnetic waves
In electromagnetic waves, the electric field \( E \) and magnetic field \( B \) are related by the speed of light \( c \):
\[ E_{\text{rms}} = c B_{\text{rms}} \]
where \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \).

### Step 3: Substitute the relation into the energy density formula
Substituting \( E_{\text{rms}} = c B_{\text{rms}} \) into the electric energy density formula gives:
\[ U_E = \frac{1}{2} \epsilon_0 (c B_{\text{rms}})^2 = \frac{1}{2} \epsilon_0 c^2 B_{\text{rms}}^2 \]

### Step 4: Use the relationship between \( c \), \( \epsilon_0 \), and \( \mu_0 \)
We know that \( c^2 = \frac{1}{\mu_0 \epsilon_0} \). Therefore:
\[ U_E = \frac{1}{2} \epsilon_0 \left(\frac{1}{\mu_0 \epsilon_0}\right) B_{\text{rms}}^2 = \frac{1}{2 \mu_0} B_{\text{rms}}^2 \]

### Step 5: Conclude the relationship for magnetic energy density
From the above derivation, we see that:
\[ U_B = \frac{1}{2} \frac{B_{\text{rms}}^2}{\mu_0} \]
This shows that the energy density in the magnetic field \( U_B \) is proportional to \( B_{\text{rms}}^2 \).

### Final Answer
Thus, the energy density in the magnetic field \( B \) is proportional to \( B^2 \).

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