Stepwise Derivation of Electric Potential Due to a Dipole
The concept of electric potential due to a dipole is a vital part of understanding electrostatics in Physics. An electric dipole consists of two point charges of equal magnitude but opposite sign, separated by a specific distance. The dipole moment (p) is defined as the product of the magnitude of one charge (q) and the distance (2a) between the charges, with the direction pointing from the negative charge to the positive charge. Dipoles are important in Physics and Chemistry, where molecules and materials often behave as dipoles due to separation of charges at tiny scales.
Electric Dipole Potential: Basic Explanation
The electric potential at any point in the vicinity of a dipole can be calculated by superposing the potentials due to each charge. The potential is a scalar quantity and is given by adding the individual potentials created by each point charge of the dipole. For a dipole with charges +q and -q separated by a distance 2a, the net potential at a point P depends on its position relative to the dipole axis.
Key Formula for Electric Potential Due to a Dipole
The standard expression for electric potential (V) at a distance r from the center of the dipole, making an angle θ with the dipole axis, is:
where p = q × 2a is the dipole moment, θ is the angle between the position vector and the dipole axis, r is the distance from the dipole’s center.
Derivation: Electric Potential of a Dipole
The electric potential at a point is found by summing the potentials due to each charge:
For points far from the dipole (r >> a), and using the binomial expansion, the expression simplifies and can be written in terms of the dipole moment (p).
| Position | Electric Potential (V) | Special Notes |
|---|---|---|
| On axial line (θ = 0°) | V = (1 / 4πε0) × (p / r2) | Maximum value; along axis |
| On equatorial line (θ = 90°) | V = 0 | Potential cancels due to symmetry |
| General Point | V = (1 / 4πε0) × (p × cosθ) / r2 | Depends on angle and position |
Solving Problems: Step-by-Step Approach
- Identify dipole charges (+q, -q) and separation (2a).
- Calculate the dipole moment: p = q × 2a.
- Determine the position of the point: distance r and angle θ with respect to dipole axis.
- Apply the general formula:
V = (1 / 4πε0) × (p × cosθ) / r2 - Insert known values and solve for V.
Worked Example: Electric Potential on Dipole Axis
Suppose a dipole has a dipole moment p = 5 × 10-29 C·m and you need to find the potential at a point 10 cm from its center on the axial line.
V = (1 / 4πε0) × (p / r2)
ε0 = 8.85 × 10-12 C2N-1m-2
(1 / 4πε0) = 9 × 109 N·m2/C2
V = 9 × 109 × (5 × 10-29 / 0.01)
V = (9 × 5 × 10-29+2)
V = 45 × 10-27 V
Important Physical Concepts
- The direction of the dipole moment is always from negative to positive charge.
- The potential is highest on the axis of the dipole and zero on the equatorial line.
- A dipole’s electric field and potential diminish much faster with distance compared to a single point charge.
- In molecules like water, dipole moments explain many physical and chemical properties.
| Quantity | Formula | SI Unit |
|---|---|---|
| Dipole moment (p) | p = q × 2a | C·m |
| Electric potential (general) | V = (1 / 4πε0) × (p × cosθ) / r2 | V (Volt) |
| Potential on axial line | V = (1 / 4πε0) × (p / r2) | V |
| Potential energy in field | U = -p × E × cosθ | J (Joule) |
Applications of Dipole Potential
- Explaining behavior of polar molecules in electric fields.
- Understanding dielectric materials and capacitor design.
- Interpreting the structure of atoms and interactions in Chemistry and Biology.
| Comparison | Point Charge | Dipole |
|---|---|---|
| Formula | V = (1 / 4πε0) × (q/r) | V = (1 / 4πε0) × (p × cosθ) / r2 |
| Distance dependence | 1/r | 1/r2 |
| Angle dependence | None | cosθ |
For more on related concepts, study: Electric Dipole, Electric Potential Due to Point Charge, Potential Energy of Charges in an External Field, and Electric Field Due to Dipole.
Practice questions:
- Calculate the electric potential at a point 15 cm from the center of a dipole with dipole moment 2.5 × 10-29 C·m on its axial line.
- What would be the potential at the same distance along the equatorial line?
For clarity on more Physics concepts, continue exploring Vedantu’s Physics resources and deepen your understanding step by step.
FAQs on Electric Potential Due to an Electric Dipole Explained
1. What is electric potential due to a dipole?
Electric potential due to a dipole is the work done in bringing a unit positive charge from infinity to a specific point in the field created by an electric dipole. It depends on the dipole moment (p), the distance to the point (r), and the angle (θ) between the dipole axis and the line joining the dipole’s center to the point. The formula is:
V = (1 / 4πε0) × (p · cosθ) / r2.
2. What is the formula for electric potential due to a dipole?
The electric potential (V) due to a dipole at a point is:
V = (1 / 4πε0) × (p · cosθ) / r2
Where:
- p = dipole moment
- θ = angle between dipole axis and line joining center to the point
- r = distance from dipole center
- ε0 = permittivity of free space
3. How do you derive the electric potential due to a dipole?
The electric potential due to a dipole is derived by:
- Writing the potentials due to +q and -q at a point P.
- Applying the binomial approximation (for r ≫ a).
- Combining both potentials algebraically.
- Expressing the result in terms of the dipole moment p = q × 2a.
V = (1 / 4πε0) × (p · cosθ) / r2
4. What is the electric potential on the axial axis of a dipole?
On the axial axis (θ = 0°), the electric potential is:
V = (1 / 4πε0) × (2p / r2).
Key Points:
- Maximum potential occurs on the axis.
- Direction from negative to positive charge.
- This is frequently asked in exams.
5. What is the electric potential at a point on the equatorial line of a dipole?
On the equatorial (perpendicular bisector) line (θ = 90°),
V = 0 exactly at the center, but off-center it becomes:
V = (1 / 4πε0) × (-p / r2).
Key Features:
- The potential is negative (opposite sign to axial).
- Result arises due to opposite contributions from +q and -q.
6. What is the difference between electric field and electric potential for a dipole?
Electric field describes the force experienced by a unit positive charge, while electric potential describes the work done in bringing a unit charge to a point.
Key differences:
- Field (E): Vector, direction matters, dependency: E ∝ 1/r3 for a dipole.
- Potential (V): Scalar, only magnitude and sign, dependency: V ∝ 1/r2.
- Potential relates to field by: E = -∇V (gradient of potential)
7. How do you calculate the electric dipole moment?
The electric dipole moment (p) is given by:
p = q × 2a
Where:
- q = magnitude of each charge (one positive, one negative)
- 2a = distance between the charges
8. What is the SI unit of electric potential and dipole moment?
The SI unit of electric potential is Volt (V).
The SI unit of electric dipole moment is Coulomb-meter (C·m).
9. What is the potential energy of a dipole in an external electric field?
The potential energy (U) of a dipole in a uniform electric field (E) is:
U = -p · E · cosθ
Where:
- p = dipole moment
- E = external electric field strength
- θ = angle between p and E
10. Is the electric potential maximum on the dipole axis?
Yes, the electric potential is maximum on the dipole axis (axial line, θ = 0°), because cosθ = 1 and the formula becomes:
V = (1 / 4πε0) × (p / r2)
This is the greatest possible value for a given dipole and distance.
11. How does the electric potential due to a point charge differ from that of a dipole?
For a point charge: V = (1 / 4πε0) × (q / r)
For a dipole: V = (1 / 4πε0) × (p · cosθ) / r2
- Point charge potential falls off as 1/r
- Dipole potential falls off faster, as 1/r2
- Dipole potential also depends on the angle θ
12. How does the electric potential of a dipole change with distance?
The electric potential due to a dipole decreases with the square of the distance (r2) from the center of the dipole. This means:
- As you move farther away (increase r), the potential drops rapidly.
- This contrasts with the potential from a single point charge, which decreases as 1/r.
