Electrostatics

We need to find the correct expression for the electric potential at a point on the axial line of an electric dipole with dipole moment and length , at a distance from the dipole’s center, and select the correct option from: , , , and .

1. Electric Dipole Configuration:
An electric dipole consists of two charges, and , separated by a distance . The dipole moment is , directed from the negative to the positive charge. The axial point lies along the dipole axis at a distance from the center of the dipole.

2. Electric Potential Formula:
The electric potential due to a point charge at distance is:


where is the permittivity of free space. The total potential is the sum of contributions from both charges.

3. Positioning the Charges:
Place the dipole along the x-axis, centered at the origin, with at and at . For a point on the axial line at (where ), calculate the distances:
- Distance from (at ) to the point: .
- Distance from (at ) to the point: .

4. Calculating the Potential:
The potential due to the positive charge:


The potential due to the negative charge:


The total potential is:

5. Simplifying the Expression:
Factor out common terms:


Combine the fractions:


Thus:


Since , we get:

6. Approximation for Large :
In many physics problems, , so is negligible compared to :


Thus:

7. Comparing with Given Options:
The exact expression is , but none of the options include the factor , suggesting the problem expects the potential without this constant (common in some contexts where constants are omitted for simplicity) or assumes the far-field approximation. Let’s evaluate the options:
- : Matches the approximate form for .
- : Does not match dimensionally or mathematically, as it gives units of , incorrect for potential.
- : Matches the denominator structure but has a squared term, leading to incorrect units ( ) and magnitude.
- : Yields units of , incorrect for potential, and lacks dependence on .

8. Selecting the Correct Option:
The option corresponds to the standard far-field approximation (without the constant, as per the options provided). The exact form is not listed, and the other options are dimensionally or mathematically inconsistent.

Final Answer:
The correct expression for the electric potential on the axial point of the dipole, assuming the far-field approximation ( ), is:

We need to determine the correct relationship between the electric field and the potential difference between two points and , with potentials and , separated by distance , as shown in the diagram where the electric field points from to . The options are: A) , B) , C) , D) .

1. Understanding Electric Potential and Field:
The electric field is related to the electric potential by the equation:


where is the potential difference over a distance , and the negative sign indicates that the field points in the direction of decreasing potential.

2. Analyzing the Diagram:
The electric field points from point to point , meaning is at a higher potential than (since the field points from higher to lower potential). Thus, , and the potential difference is:


The distance between and is given as .

3. Relating Potential Difference to Electric Field:
For a uniform electric field (assumed here, as the field lines are parallel), the potential difference between two points is:


Since the field points from to , is the initial point (higher potential) and is the final point (lower potential):

4. Evaluating the Options:
- Option A: . This is dimensionally incorrect (units of , not potential) and does not match the formula.
- Option B: . This matches our derived expression and has correct units ( ).
- Option C: . Since , is negative, but is positive, so this is incorrect.
- Option D: . The left-hand side is zero, and the right-hand side is nonzero with incorrect units ( ), so this is incorrect.

Final Answer:
The correct relationship between the electric field and the potential difference is: So, the correct option is A.