Statement I: If electric potential in certain region is constant, then the electric field must be zero in this region.
Statement II: `vecE = -(dV)/(dr) hatv`.

Options

A. Statement I is true, Statement II is True, Statement II is a correct explanation for statement I.
B. Statement I is true, Statement II is True, Statement II is Not a correct explanation for statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Explanation

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
**Statement I:** If electric potential in a certain region is constant, then the electric field must be zero in this region.

- **Explanation:** Electric potential (V) is defined as the work done per unit charge in bringing a charge from a reference point to a specific point in an electric field without any acceleration. If the electric potential is constant in a region, it means that there is no change in potential energy as you move through that region. Therefore, the electric field, which is the negative gradient of the electric potential, must be zero. Mathematically, if V is constant, then:
\[ \frac{dV}{dr} = 0 \implies \vec{E} = -\frac{dV}{dr} \hat{v} = 0 \]
Thus, Statement I is true.

### Step 2: Analyze Statement II
**Statement II:** \(\vec{E} = -\frac{dV}{dr} \hat{v}\).

- **Explanation:** This statement is a fundamental definition of the electric field. The electric field (E) is indeed defined as the negative rate of change of electric potential (V) with respect to position (r). This relationship holds true in electrostatics and is derived from the concept of potential difference and work done in an electric field. Therefore, Statement II is also true.

### Step 3: Conclusion
Since both statements are true, we need to determine if Statement II provides a correct explanation for Statement I.

- **Explanation of Relationship:** The relationship defined in Statement II (\(\vec{E} = -\frac{dV}{dr} \hat{v}\)) explains why if the potential is constant (as stated in Statement I), the electric field must be zero. If there is no change in potential (constant V), then the derivative (rate of change) is zero, leading to an electric field of zero.

### Final Answer
Both statements are true, and Statement II is the correct explanation for Statement I.

Thus, the answer is: **Both Statement I and Statement II are true, and Statement II is the correct explanation for Statement I.**

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