Question

Consider an electrostatic field \(\overrightarrow{E}\) in a region of space. Identify the INCORRECT statement.

• A. The work done in moving a charge in a closed path inside the region is zero
• B. The curl of \(\overrightarrow{E}\) is zero
• C. The field can be expressed as the gradient of a scalar potential
• D. The potential difference between any two points in the region is always zero

Answer 1

The Correct Option is D

The question pertains to the properties of an electrostatic field \(\overrightarrow{E}\). We are tasked with identifying the incorrect statement among the given options. Let's analyze each one:

1. The work done in moving a charge in a closed path inside the region is zero:
   In an electrostatic field, it is a characteristic that the work done along a closed path is always zero. This is because electrostatic forces are conservative forces. Therefore, this statement is correct.
2. The curl of \(\overrightarrow{E}\) is zero:
   The curl of an electrostatic field is always zero because electrostatic fields are derived from scalar potentials. Mathematically, this is expressed as \(\nabla \times \overrightarrow{E} = 0\). Thus, this statement is also correct.
3. The field can be expressed as the gradient of a scalar potential:
   An electrostatic field can indeed be expressed as the negative gradient of a scalar potential, i.e., \(\overrightarrow{E} = -\nabla V\). Hence, this statement is true.
4. The potential difference between any two points in the region is always zero:
   This statement is incorrect. In an electrostatic field, the potential difference between two points is generally non-zero unless those two points are at the same potential. The potential difference depends on the potential values at those points. Therefore, this statement is the incorrect one.

Thus, the provided statement "The potential difference between any two points in the region is always zero" is indeed the incorrect statement in the context of an electrostatic field.

Conclusion:
The correct answer is that the potential difference between any two points in the region is not always zero, and it is the incorrect statement as per the properties of an electrostatic field.