The maxwells four equations are written as
i. `oint vec(E).vec(dS) = (q)/(epsilon_(0))`
ii.`oint vec(B).vec(dS) = 0 `
iii. `oint vec(E ).vec(dl) = (d)/(dt) oint vec(B).vec(dS)`
iv. `oint vec(B).vec(dl) = mu_(0) epsilon (d)/(dt) oint vec(E).vec(dS)`
The equations which have sources of `vec(E ) ` and `vec(B)` are

Options

A. (i), (ii),(iii)
B. (i), (ii)
C. (i) and (iii) only
D. (i) and (iv) only

Answer

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

### Step-by-Step Solution:

1. **Identify the First Equation:**
\[ \oint \vec{E} \cdot \vec{dS} = \frac{q}{\epsilon_0} \]
This equation is known as Gauss's Law for electricity. It states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Hence, this equation has a source: the electric field \(\vec{E}\) has sources in the form of electric charge \(q\).

2. **Identify the Second Equation:**
\[ \oint \vec{B} \cdot \vec{dS} = 0 \]
This equation is Gauss's Law for magnetism. It indicates that there are no magnetic monopoles; the magnetic flux through a closed surface is zero. Therefore, this equation does not have a source for the magnetic field \(\vec{B}\).

3. **Identify the Third Equation:**
\[ \oint \vec{E} \cdot \vec{dl} = \frac{d}{dt} \oint \vec{B} \cdot \vec{dS} \]
This is Faraday's Law of Induction. It relates the electric field around a closed loop to the rate of change of magnetic flux through the loop. While it describes how changing magnetic fields can induce electric fields, it does not have a source term for \(\vec{E}\) or \(\vec{B}\).

4. **Identify the Fourth Equation:**
\[ \oint \vec{B} \cdot \vec{dl} = \mu_0 \epsilon_0 \frac{d}{dt} \oint \vec{E} \cdot \vec{dS} \]
This equation is known as the Ampère-Maxwell Law. It relates the magnetic field around a closed loop to the rate of change of electric flux through the loop. Similar to the third equation, it does not contain a source term for \(\vec{B}\) or \(\vec{E}\).

### Conclusion:
- The equations that have sources are:
- **First Equation:** \(\oint \vec{E} \cdot \vec{dS} = \frac{q}{\epsilon_0}\) (source of \(\vec{E}\))

- The other equations (Second, Third, and Fourth) do not have sources for \(\vec{E}\) or \(\vec{B}\).