At a point on the axis of an electric dipole

Options

A. the electric field E is zero
B. the electric potential V is zero
C. both E and V are zero
D. neither E nor V is zero

Answer

To solve the problem regarding the electric field and electric potential at a point on the axis of an electric dipole, we can follow these steps:

### Step 1: Understanding the Electric Dipole
An electric dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance of 2L. The dipole moment \( p \) is defined as:
\[ p = Q \cdot 2L \]

### Step 2: Determine the Position
Let’s consider a point P located at a distance R from the center of the dipole along its axis. The distance from the positive charge (+Q) to point P is \( R - L \), and the distance from the negative charge (-Q) to point P is \( R + L \).

### Step 3: Calculate the Electric Potential (V)
The electric potential \( V \) at point P due to the dipole is given by the formula:
\[ V = k \left( \frac{Q}{R - L} - \frac{Q}{R + L} \right) \]
where \( k \) is the Coulomb's constant.

To simplify this, we can find a common denominator:
\[ V = kQ \left( \frac{(R + L) - (R - L)}{(R - L)(R + L)} \right) \]
This simplifies to:
\[ V = kQ \left( \frac{2L}{R^2 - L^2} \right) \]

### Step 4: Analyze the Limit as R becomes much larger than L
If \( R \) is much larger than \( L \) (i.e., \( R \gg L \)), we can neglect \( L^2 \) in the denominator:
\[ V \approx \frac{2kQ \cdot L}{R^2} = \frac{kp}{R^2} \]
This indicates that the electric potential is not zero at point P.

### Step 5: Calculate the Electric Field (E)
The electric field \( E \) at point P due to the dipole can be calculated using:
\[ E = -\frac{dV}{dR} \]
Calculating the derivative of \( V \):
\[ E = -\frac{d}{dR} \left( \frac{kp}{R^2} \right) = \frac{2kp}{R^3} \]
This shows that the electric field is also not zero at point P.

### Step 6: Conclusion
From the calculations:
- The electric potential \( V \) at point P is not zero.
- The electric field \( E \) at point P is also not zero.

Thus, the correct answer is that neither the electric field nor the electric potential is zero at a point on the axis of an electric dipole.