Author Jonathan David
Author Jonathan David
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Electric Dipoles

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[21.7] Electric Dipoles: Point charges q1 = -4.5 nC and q2 = +4.5 nC are separated by 3.1 mm, …

Point charges q1 = -4.5 nC and q2 = +4.5 nC are separated by 3.1 mm, forming an electric dipole. [a] Find the electric dipole moment (magnitude and direction). [b] The charges are in a uniform electric field whose direction makes an angle of 36.9° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2 × 10⁻⁹ N·m?

Reference: Young, H. D., Freedman, R. A., & Ford, A. L. (2020). University Physics with Modern Physics (Most Editions). Pearson. Online Purchase https://amzn.to/4f6yCuY

[21.7] Electric Dipoles

Point charges q_1 = -4.5 \, \mathrm{nC} and q_2 = +4.5 \, \mathrm{nC} are separated by 3.1 \, \mathrm{mm}, forming an electric dipole.

[a] Find the electric dipole moment (magnitude and direction).

[b] The charges are in a uniform electric field whose direction makes an angle of 36.9^\circ with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m}?

[a][Step 1] List all given data, unknown data, related formulae and convert units.

  • q_1 = -4.5 \, \mathrm{nC} = -4.5 \times 10^{-9} \, \mathrm{C}
  • q_2 = +4.5 \, \mathrm{nC} = +4.5 \times 10^{-9} \, \mathrm{C}
  • r = 3.1 \, \mathrm{mm} = 3.1 \times 10^{-3} \, \mathrm{m}

Magnitude of electric dipole moment:

p = q \cdot d = (4.5 \times 10^{-9} \, \mathrm{C})(3.1 \times 10^{-3} \, \mathrm{m}) = 1.4 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m}

[a][Step 2]

Take Note:

  1. An electric dipole consists of two equal and opposite charges separated by a distance. The direction of the electric dipole moment is conventionally defined as pointing from the negative charge to the positive charge.
  2. The direction of the electric field and the force in electrostatics is conventionally defined to go from positive to negative charge for several reasons.
  3. An electric dipole is a pair of point charges with equal magnitude and opposite sign.

[b][Step 1]

Magnitude of the net torque:

  • \phi = 36.9^\circ
  • \tau = 7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m}

\tau = (qE)(d \sin \phi)

[b][Step 2] Assemble and execute.

This is not a difficult question by any means, but as always, it is the detail that can throw us off, such as the angle. Do we use the provided angle or alter it?

In this case, we can look at the electric field as the +x-axis.

    \[ E = \frac{\tau}{qd \sin \phi} = \frac{7.2 \times 10^{-9} \, \mathrm{N} \cdot \mathrm{m}}{(1.4 \times 10^{-11} \, \mathrm{C} \cdot \mathrm{m}) \sin(180^\circ - 36.9^\circ)} \approx 856.543 \, \mathrm{N/C} \]

Therefore, E = 8.6 \times 10^2 \, \mathrm{N/C}.

Electric Dipoles

Electric Dipoles

What is an Electric Dipole?

An electric dipole is a system of two equal but opposite charges separated by a fixed distance. It is an essential concept in electrostatics and helps explain the behavior of molecules and charges in external electric fields.

A common example of an electric dipole is a water molecule, where the distribution of charges creates a dipole moment.

Dipole Moment

The strength of an electric dipole is characterized by its dipole moment \vec{p}, which is defined as:

    \[\vec{p} = q \vec{d}\]

Here:

  • q: Magnitude of one of the charges (in coulombs, \text{C}).
  • \vec{d}: Vector pointing from the negative charge to the positive charge (in meters, \text{m}).

The unit of the dipole moment is \text{C·m} (coulomb-meters).

Electric Field Due to a Dipole

The electric field created by a dipole depends on the location where it is measured:

  • Along the axial line: The electric field at a distance r from the center of the dipole is:

        \[E_{\text{axial}} = \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^3}\]

  • Along the equatorial line: The electric field at a distance r from the center of the dipole is:

        \[E_{\text{equatorial}} = \frac{1}{4 \pi \epsilon_0} \frac{p}{r^3}\]

Here:

  • p: Dipole moment.
  • \epsilon_0: Permittivity of free space (8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2).

Torque on a Dipole in a Uniform Electric Field

When a dipole is placed in a uniform electric field \vec{E}, it experiences a torque that tends to align the dipole with the field. The torque \tau is given by:

    \[\tau = \vec{p} \times \vec{E}\]

The magnitude of the torque is:

    \[\tau = pE \sin\theta\]

Where \theta is the angle between \vec{p} and \vec{E}.

Potential Energy of a Dipole in a Uniform Electric Field

The potential energy U of a dipole in a uniform electric field is given by:

    \[U = -\vec{p} \cdot \vec{E}\]

The magnitude of the potential energy is:

    \[U = -pE \cos\theta\]

The dipole is in stable equilibrium when \theta = 0^\circ (aligned with the field) and in unstable equilibrium when \theta = 180^\circ (opposite to the field).

Applications of Electric Dipoles

Electric dipoles have numerous applications, including:

  • Explaining the behavior of polar molecules in electric fields.
  • Designing antennas for communication systems.
  • Understanding the electric properties of materials.

Key Takeaways

Electric dipoles are fundamental systems in electrostatics that help describe interactions between charges and fields. Their behavior under electric fields is crucial in understanding molecular properties, electrical engineering, and materials science.