Which of the following Maxwell's equations have sources of `vecE` and `vecB`?

Options

A. `oint_SvecE. vec(ds)=q/(in_0)`
B. `oint_SvecB. vec(dl)=mu_0I+mu_0in_0d/(dt) oint_SvecE.vec(ds)`
C. `oint_SvecE. vec(dl)=-d/(dt)oint_S vecB.vec(ds)`
D. `oint_SvecB. vec(ds)=0`

Explanation

To determine which of Maxwell's equations have sources of the electric field \(\vec{E}\) and the magnetic field \(\vec{B}\), we can analyze each of the four equations.

### Step-by-Step Solution:

1. **Understanding Maxwell's Equations**:
Maxwell's equations consist of four fundamental equations that describe how electric and magnetic fields interact. They can be expressed in both integral and differential forms.

2. **Identifying the Equations**:
The four Maxwell's equations in integral form are:
- **Gauss's Law for Electricity**:
\[ \oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0} \]
- **Gauss's Law for Magnetism**:
\[ \oint \vec{B} \cdot d\vec{A} = 0 \]
- **Faraday's Law of Induction**:
\[ \oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \]
- **Ampère-Maxwell Law**:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} + \mu_0 I_{\text{enc}} \]

3. **Analyzing Each Equation for Sources**:
- **Gauss's Law for Electricity**: This equation has a source term, \(Q\) (the total charge enclosed), which is the source of the electric field \(\vec{E}\).
- **Gauss's Law for Magnetism**: This equation states that there are no magnetic monopoles; hence, it does not have a source for \(\vec{B}\).
- **Faraday's Law of Induction**: This equation describes how a changing magnetic field induces an electric field, but it does not have a source term for \(\vec{E}\).
- **Ampère-Maxwell Law**: This equation includes a source term, \(I_{\text{enc}}\) (the enclosed current), which is a source for the magnetic field \(\vec{B}\).

4. **Conclusion**:
- The equations that have sources are:
- **Gauss's Law for Electricity** (source of \(\vec{E}\))
- **Ampère-Maxwell Law** (source of \(\vec{B}\))

### Final Answer:
- **Maxwell's equations that have sources**:
- Gauss's Law for Electricity (source of \(\vec{E}\))
- Ampère-Maxwell Law (source of \(\vec{B}\))