The maxwells four equations are written as
(`i`) `ointvecE.vec(dS)=(q_(0))/(epsilon_(0))`
(`ii`) `ointvecB.vec(dS)=0`
(`iii`) `ointvecE.vec(dl)=(d)/(dt)ointvecB.vec(dS)`
(`iv`) `ointvecB.vec(dl)=mu_(0)epsilon_(0)(d)/(dt)ointvecE.vec(dS)`
The equations which have sources of `vecE` and `vecB` 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 equation step by step.

### Step 1: Analyze Equation (i)
The first equation is given by:
\[ \oint \vec{E} \cdot d\vec{S} = \frac{q_0}{\epsilon_0} \]
This equation is Gauss's law for electricity. It states that the electric flux through a closed surface is proportional to the charge enclosed within that surface. Thus, this equation has a source for the electric field \(\vec{E}\) (the charge \(q_0\)).

### Step 2: Analyze Equation (ii)
The second equation is:
\[ \oint \vec{B} \cdot d\vec{S} = 0 \]
This equation indicates that the magnetic flux through a closed surface is zero, implying that there are no magnetic monopoles (sources) in classical electromagnetism. Therefore, this equation does not have a source for the magnetic field \(\vec{B}\).

### Step 3: Analyze Equation (iii)
The third equation is:
\[ \oint \vec{E} \cdot d\vec{l} = \frac{d}{dt} \oint \vec{B} \cdot d\vec{S} \]
This is Faraday's law of induction, which relates the electric field around a closed loop to the rate of change of magnetic flux through the loop. While it describes how an electric field can be induced by a changing magnetic field, it does not have a source for the electric field \(\vec{E}\).

### Step 4: Analyze Equation (iv)
The fourth equation is:
\[ \oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d}{dt} \oint \vec{E} \cdot d\vec{S} \]
This is Ampère-Maxwell law, which states that the line integral of the magnetic field around a closed loop is related to the rate of change of electric flux through the loop and the current. However, it does not have a source for the magnetic field \(\vec{B}\) in the classical sense.

### Conclusion
From the analysis:
- Equation (i) has a source for \(\vec{E}\) (the charge \(q_0\)).
- Equation (ii) does not have a source for \(\vec{B}\).
- Equation (iii) does not have a source for \(\vec{E}\).
- Equation (iv) does not have a source for \(\vec{B}\).

Thus, the equations which have sources are:
- **(i)** for \(\vec{E}\).

### Final Answer
The equation which has a source of \(\vec{E}\) is (i). There are no equations with sources for \(\vec{B}\).

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