Dipole
Understanding the Concept of a Dipole
In the field of electrostatics, a dipole is a system of two equal but opposite charges or magnetic poles separated by a small distance. Dipoles are important in various areas of physics and chemistry, including molecular chemistry, where the concept explains the behavior of polar molecules.
Electric Dipole
An electric dipole consists of two charges, ( +q ) and ( -q ), separated by a distance ( 2a ), where ( a ) is the distance from the center to each charge.
Dipole Moment
The dipole moment (( \vec{p} )) is a vector quantity defined as the product of the charge ( q ) and the separation distance ( 2a ). It is directed from the negative charge to the positive charge.
[ \vec{p} = q \cdot (2a) \cdot \hat{r} ]
where ( \hat{r} ) is the unit vector in the direction from the negative to the positive charge.
Electric Field Due to a Dipole
The electric field (( \vec{E} )) due to a dipole at a point in space depends on the position of the point relative to the dipole. For points along the axial line (the line extending through both charges), the electric field is given by:
[ \vec{E}_{\text{axial}} = \frac{1}{4\pi\epsilon_0} \cdot \frac{2p}{r^3} ]
For points along the equatorial line (the line perpendicular to the axial line and passing through the center of the dipole), the electric field is:
[ \vec{E}_{\text{equatorial}} = -\frac{1}{4\pi\epsilon_0} \cdot \frac{p}{r^3} ]
where ( \epsilon_0 ) is the vacuum permittivity, ( p ) is the magnitude of the dipole moment, and ( r ) is the distance from the center of the dipole to the point.
Potential Energy of a Dipole in an External Field
When an electric dipole is placed in an external electric field (( \vec{E} )), it experiences a torque (( \vec{\tau} )) that tends to align the dipole with the field. The potential energy (( U )) of the dipole in the field is given by:
[ U = -\vec{p} \cdot \vec{E} ]
Torque on a Dipole in an External Field
The torque (( \vec{\tau} )) experienced by a dipole in an external electric field is given by the cross product of the dipole moment and the electric field:
[ \vec{\tau} = \vec{p} \times \vec{E} ]
Magnetic Dipole
A magnetic dipole is analogous to an electric dipole but consists of a loop of current or a pair of north and south magnetic poles. The magnetic dipole moment (( \vec{m} )) is a vector quantity that represents the strength and orientation of the magnetic dipole.
Magnetic Dipole Moment
For a current loop, the magnetic dipole moment is given by the product of the current (( I )) and the area of the loop (( A )):
[ \vec{m} = I \cdot A \cdot \hat{n} ]
where ( \hat{n} ) is the unit vector normal to the plane of the loop.
Magnetic Field Due to a Dipole
The magnetic field (( \vec{B} )) due to a magnetic dipole can be calculated using the Biot-Savart law or Ampère's circuital law. For points far from the dipole, the field resembles that of an electric dipole.
Differences Between Electric and Magnetic Dipoles
| Property | Electric Dipole | Magnetic Dipole |
|---|---|---|
| Constituents | Two opposite charges (+q and -q) | Current loop or north and south poles |
| Dipole Moment | ( \vec{p} = q \cdot (2a) \cdot \hat{r} ) | ( \vec{m} = I \cdot A \cdot \hat{n} ) |
| Field Calculation | Coulomb's law | Biot-Savart law or Ampère's circuital law |
| Torque | ( \vec{\tau} = \vec{p} \times \vec{E} ) | ( \vec{\tau} = \vec{m} \times \vec{B} ) |
| Potential Energy | ( U = -\vec{p} \cdot \vec{E} ) | ( U = -\vec{m} \cdot \vec{B} ) |
| Field at Large Dist. | ( \vec{E} \propto \frac{1}{r^3} ) | ( \vec{B} \propto \frac{1}{r^3} ) |
Examples
Example 1: Electric Dipole
Consider an electric dipole with charges ( +q ) and ( -q ) separated by a distance of ( 2a = 2 \times 10^{-10} ) m. If ( q = 1.6 \times 10^{-19} ) C, the dipole moment is:
[ \vec{p} = q \cdot (2a) = (1.6 \times 10^{-19} \text{ C}) \cdot (2 \times 10^{-10} \text{ m}) = 3.2 \times 10^{-29} \text{ C}\cdot\text{m} ]
Example 2: Magnetic Dipole
A circular loop of wire with a radius of ( 0.05 ) m carries a current of ( 2 ) A. The magnetic dipole moment is:
[ \vec{m} = I \cdot A = I \cdot \pi r^2 = (2 \text{ A}) \cdot \pi \cdot (0.05 \text{ m})^2 \approx 0.0157 \text{ A}\cdot\text{m}^2 ]
In summary, dipoles are fundamental concepts in electrostatics and magnetostatics, with wide-ranging applications in physics and chemistry. Understanding their properties, such as the dipole moment, field distributions, and interactions with external fields, is crucial for studying molecular structures, electromagnetic theory, and material science.